Hyperbolic Manifolds and Discrete Groups

Michael Kapovich August 23, 1998

Contents

1 Introduction 1 1.1 Big Monster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Overview of the Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Beyond the Hyperbolization Theorem . . . . . . . . . . . . . . . . . . . . . 5

2 3-dimensional Topology 8 2.1 Basics of basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Incompressible surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Existence theorems for superincompressible surfaces . . . . . . . . . . . . . 12 2.4 Deforming homotopy-equivalences to homeomorphisms . . . . . . . . . . . . 14 2.5 Paired 3-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.6 Seifert manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.7 Decomposition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.8 Characteristic submanifolds in 3-manifolds . . . . . . . . . . . . . . . . . . . 17 2.9 3-manifolds fibered over 𝕊^{1} . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.10 Baumslag-Solitar relations in 3-manifold groups . . . . . . . . . . . . . . . . 20

3 Thurston’s Norm and Incompressible Surfaces 22 3.1 Norms defined over ℤ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Variation of fiber-bundle structure . . . . . . . . . . . . . . . . . . . . . . . 23 3.3 Thurston’s norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Geometry of the Hyperbolic Space 28 4.1 General definitions and notations . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2 CAT (λ)-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.3 The basic properties of ℍ^{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.4 Models of the hyperbolic space . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.5 Isometries of the hyperbolic space . . . . . . . . . . . . . . . . . . . . . . . . 33 4.6 Convergence property for Moebius transformations . . . . . . . . . . . . . . 35 4.7 Convex polyhedra in the hyperbolic space . . . . . . . . . . . . . . . . . . . 36 4.8 Cayley graphs of finitely generated groups . . . . . . . . . . . . . . . . . . . 37 4.9 Quasi-isometries and quasi-conformal homeomorphisms . . . . . . . . . . . 37 4.10 Length distortion under quasiconformal maps of the hyperbolic space …. 41 4.11 Harmonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5 Teichmüller Theory of Riemann surfaces 43 5.1 Properties of quasiconformal maps . . . . . . . . . . . . . . . . . . . . . . . 43 5.2 Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.3 Geometry of quadratic differentials . . . . . . . . . . . . . . . . . . . . . . . 45 5.4 Teichmüller spaces of Riemann surfaces . . . . . . . . . . . . . . . . . . . . 46 5.5 Locally finite subgroups of Mod_{S} . . . . . . . . . . . . . . . . . . . . . . . . 49

6 Kleinian Groups 51 6.1 Representation varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.2 Residual finiteness and Selberg lemma . . . . . . . . . . . . . . . . . . . . . 53 6.3 Basics of discrete groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.4 Elementary groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.5 Properties of limit sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.6 Quotient spaces of discrete groups . . . . . . . . . . . . . . . . . . . . . . . 56 6.7 Fundamental domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.8 Poincare theorem on fundamental polyhedra . . . . . . . . . . . . . . . . . . 57 6.9 Geometrically finite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.10 Kleinian groups and Riemann surfaces . . . . . . . . . . . . . . . . . . . . . 62 6.11 Kazhdan-Margulis-Zassenhaus lemma . . . . . . . . . . . . . . . . . . . . . 63 6.12 Geometric consequences of Kazhdan-Margulis-Zassenhaus lemma . . . . . . 65 6.13 Characterization of geometrically finite groups . . . . . . . . . . . . . . . . 68 6.14 Convex hull and the domain of discontinuity . . . . . . . . . . . . . . . . . . 69 6.15 Combination Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.15.1 Klein Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.15.2 Maskit Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.16 Ahlfors finiteness theorem and its companions . . . . . . . . . . . . . . . . . 78 6.17 Extensions of the Ahlfors Finiteness Theorem . . . . . . . . . . . . . . . . . 79 6.18 Limit sets of geometrically finite groups . . . . . . . . . . . . . . . . . . . . 80 6.19 Supgroups of finitely generated Kleinian groups . . . . . . . . . . . . . . . . 80 6.20 Ends of hyperbolic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7 Introduction to orbifold theory 83 7.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.2 2-dimensional orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.3 3-dimensional locally reflective orbifolds . . . . . . . . . . . . . . . . . . . . 90 7.4 Orbifolds of finite type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.5 A homeomorphism theorem for a class of 3-orbifolds . . . . . . . . . . . . . 97

8 Complex Projective Structures 101 8.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 8.2 The holonomy theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 8.3 Grafting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

9 Sociology of Kleinian Groups 105 9.1 Algebraic convergence of representations . . . . . . . . . . . . . . . . . . . . 105 9.2 Geometric convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 9.3 Isomorphisms of geometrically finite groups . . . . . . . . . . . . . . . . . . 109 9.4 Mostow Rigidity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 9.5 Sullivan Rigidity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 9.6 Bers isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 9.7 Smoothness of the representation variety . . . . . . . . . . . . . . . . . . . . 115 9.8 Calculation of dimension of the representation variety . . . . . . . . . . . . 116 9.9 Space of quasi-fuchsian representations . . . . . . . . . . . . . . . . . . . . . 117 9.10 Distortion of the translational length under quasiconformal conjugation . . 118 9.11 Proof of the Ahlfors Recurrence Theorem . . . . . . . . . . . . . . . . . . . 118 9.12 A generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 9.13 Totally degenerate groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 9.14 Algebraic topology versus geometric topology . . . . . . . . . . . . . . . . . 121 9.15 Geometric profile of the algebraic convergence . . . . . . . . . . . . . . . . . 122

10 Ultralimits of Metric Spaces 125 10.1 Ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 10.2 Ultralimits of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 10.3 The asymptotic cone of a metric space . . . . . . . . . . . . . . . . . . . . . 128 11 Actions of Groups on Trees 129

11.1 Basic definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . 129 11.2 Actions on simplicial trees, graphs of groups and graphs of spaces . . . . . . 131 11.3 Equivariant Gromov–Hausdorff convergence . . . . . . . . . . . . . . . . . . 134 11.4 Limits of isometric actions on CAT (0)-spaces . . . . . . . . . . . . . . . . . 134 11.5 Representation varieties and actions of groups on trees . . . . . . . . . . . . 135 11.6 Proper actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

12 Measured Laminations, Foliations and Actions of Surface Groups on Trees 138 12.1 Euclidean motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 12.2 Geodesic currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 12.3 Measured foliations on hyperbolic surfaces . . . . . . . . . . . . . . . . . . . 140 12.4 Train tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 12.5 Measured geodesic laminations . . . . . . . . . . . . . . . . . . . . . . . . . 146 12.6 Train tracks and convergence of measured laminations . . . . . . . . . . . . 151 12.7 From measured foliations to measured laminations . . . . . . . . . . . . . . 152 12.8 From laminations to foliations . . . . . . . . . . . . . . . . . . . . . . . . . . 152 12.9 Action of Mod_{S} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 12.10 The geometric intersection number . . . . . . . . . . . . . . . . . . . . . . . 154 12.11 From laminations to trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 12.12 Interval exchange transformations . . . . . . . . . . . . . . . . . . . . . . . 156 12.13 An application of Skora’s theorem . . . . . . . . . . . . . . . . . . . . . . . 157 12.14 Characterization of aperiodic homeomorphisms . . . . . . . . . . . . . . . . 158 12.15 Dynamics of pseudo-Anosov homeomorphisms . . . . . . . . . . . . . . . . 160 12.16 Thurston’s compactification of the Teichmüller space . . . . . . . . . . . . 162

13 Singular Hyperbolic Metrics and Pleated Surfaces 164 13.1 Singular hyperbolic metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 13.2 Existence theorem for singular pleated maps . . . . . . . . . . . . . . . . . . 165 13.3 Compactness theorem for pleated maps into

hyperbolic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 13.4 Geometrically tame ends of hyperbolic manifolds . . . . . . . . . . . . . . . 167 13.5 Ending laminations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 13.6 Infinite coverings of hyperbolic manifolds . . . . . . . . . . . . . . . . . . . 171

14 Rips Theory 173 14.1 Stable trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 14.2 Unions of bands and band complexes . . . . . . . . . . . . . . . . . . . . . . 175 14.3 Pushing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 14.4 Transversal measure on union of bands . . . . . . . . . . . . . . . . . . . . . 180 14.5 Imanishi’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 14.6 Band complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 14.7 Holonomy of vertical paths in X . . . . . . . . . . . . . . . . . . . . . . . . 187 14.8 An analogue of Kazhdan-Margulis lemma for group actions on trees …. 188 14.9 Moves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 14.10Preliminary motions of the Rips machine . . . . . . . . . . . . . . . . . . . 195 14.11The Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 14.12Machine Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 14.13Proof of the decomposition theorem . . . . . . . . . . . . . . . . . . . . . . 206 14.14Simplicial case of Skora’s theorem . . . . . . . . . . . . . . . . . . . . . . . 208 14.15Proof of Skora’s theorem in the case of geometric group action . . . . . . . 209 14.16Geometric and nongeometric actions on trees . . . . . . . . . . . . . . . . . 212 14.17Application of Rips’ theory to compactness properties of representation va-

rieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

15 Outline of the Proof of the Hyperbolization Theorem 216 15.1 Case A: The generic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 15.2 Case B: Manifolds fibered over the circle . . . . . . . . . . . . . . . . . . . . 220

16 Reduction of the Generic Case of the Hyperbolization Theorem to the Bounded Image Theorem 221 16.1 Step 1. Maskit Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 16.2 Step 2. Formulation of the Fixed–Point Theorem . . . . . . . . . . . . . . . 221 16.3 Step 3. Proof of the Contraction Theorem . . . . . . . . . . . . . . . . . . . 222 16.4 Step 4. End of the proof of the Fixed-Point Theorem . . . . . . . . . . . . . 223

17 The Bounded Image Theorem 224 17.1 Limit exists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 17.2 Geometry of the limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

17.2.1 Preliminaries and notations . . . . . . . . . . . . . . . . . . . . . . . 225 17.2.2 Idea of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 17.2.3 Limit has no accidental parabolics . . . . . . . . . . . . . . . . . . . 229 17.2.4 Endgame: Limit is geometrically finite . . . . . . . . . . . . . . . . . 231

18 Hyperbolization of Manifolds Fibered Over 𝕊^{1} 233 18.1 Compactness theorem for aperiodic homeomorphisms . . . . . . . . . . . . . 233 18.2 The double limit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 18.3 Proof of Theorem 13.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 18.4 An Alternative Approach to Hyperbolization of Manifolds Fibered

Over 𝕊^{1}: Reduction to the Generic Case . . . . . . . . . . . . . . . . . . . . 235

19 Hyperbolization of Orbifolds 237 19.1 Hyperbolic orbifolds associated with patterns of circles . . . . . . . . . . . . 237 19.2 Brooks’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 19.3 Construction of manifold coverings over alternated orbifolds . . . . . . . . . 249 19.4 Construction of finite type hyperbolic orbifolds . . . . . . . . . . . . . . . . 250 19.5 Gluing orbifolds of the class R from orbifolds of finite type . . . . . . . . . 253 19.6 Hyperbolization of locally reflective orbifolds . . . . . . . . . . . . . . . . . 255 19.7 Hyperbolic design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

1 Introduction

This is a version of notes for my class “Hyperbolic Manifolds and Discrete Groups” which I was teaching at the University of Utah during the academic year 1993-94. I am thankful to Mladen Bestvina, Noel Brady, Steve Gersten, Andre Szabo, Domingo Toledo and Andrejs Treibergs for patience, interesting questions and suggestions. Several conversations with Mladen Bestvina were especially important. I am also grateful to Bernhard Leeb and John Millson: some of ideas of our joint works I had used in these Notes. I would like to thank Darryl Cooper who had pointed out a mistake in the original version of the Chapter 18.4.

1.1 Big Monster

Thurston’s Hyperbolization Theorem (“The Big Monster”). Suppose that M is a compact Haken atoroidal 3-manifold, whose boundary has zero Euler characteristic. Then the interior of M admits a complete hyperbolic metric of finite volume. (See the Chapter 2 for definitions.)

Thurston had announced this theorem about 20 years ago, but did not write a complete text of the proof. A proof of the “Big Monster” existed, however it was similar to an unfinished building. In these lectures I bring different pieces of the proof together and complete the “construction work” in the “generic case” of hyperbolization theorem. Initially I was planning to include detailed proof in the special case of manifolds fibered over 𝕊^{1} as well (using Skora’s theorem); however in the course of 1993-94 academic year a preliminary version of Otal’s book [162] became available, thus I will give here only a sketch of the proof in the fibered case.

J.-P. Otal wrote the books [162] (the case of manifolds fibered over 𝕊^{1}) and [163] (the generic case) with complete proof of the Hyperbolization Theorem. Unlike these Notes, the approach of [163] is analytical, it is based on McMullen’s proof of the Bounded Image Theorem.

Here are the main pieces of the proof of the Hyperbolization Theorem:

(a) The “generic case” of manifold that contains a superincompressible surface which is not 2-fold covered by a fiber in a fibration over 𝕊^{1}. The “last step of induction” of the proof in this case (the Bounded Image Theorem) has two quite different proofs. The first (analytical) proof is due to C. McMullen [140]. The second proof is essentially due to Thurston, however we replace difficult and incomplete geometric analysis of [194, 195, 196] by combinatorial arguments mostly due to Rips. We shall “reconstruct” Thurston’s proof following the outline written by J. Morgan [150] and fill gaps in his proof using the theory of group actions on trees [152], [185], [21], [171]. The orbifold-trick reduction of the general step of induction to the last one is described by Morgan in [150]. There are several details missing in this part of the proof: a generalized Andreev’s theorem and a homeomorphism theorem for orbifolds with isomorphic fundamental groups. To fill the first gap I will use a Theorem of Brooks [32], alternatively one can use a generalized Andreev’s Theorem [192], as it is done in Otal’s book [163].

(b) The “exceptional case”, when M is a surface bundle over 𝕊^{1}, was originally treated by Thurston in [195]. Unfortunately Thurston’s proof was incomplete: parts of it rely upon forthcoming Thurston’s preprints which had failed to materialize. An alternative proof was given by Otal in [162], who used actions of groups on trees. Another variation on this proof see in [141]. In the Chapter 18 I will outline Otal’s proof (I omitted only one, but central, ingredient: estimates on the length function). I decided to include this outline in my notes since it illustrates the power of the technique based on studying group actions on trees. In the same Chapter 18 I give an alternative proof of the Hyperbolization Theorem for manifolds fibered over the circle assuming existence of a finite covering with the 1-st Betti number ≥ 2.^{1}

1.2 Overview of the Notes

Here is the basic idea behind the proof of Thurston’s Hyperbolization Theorem. Assume that M is a closed atoroidal orientable Haken manifold, S ⊂ M is an incompressible surface, which separates M into components M_{1}, M_{2} and each M_{i} admits a complete hyperbolic metric. These metrics correspond to holonomy representations ρ_{i} : π_{1}(M_{i}) → P SL(2, ℂ). The goal is to deform hyperbolic structures on M_{i} so that the representations ρ_{1}, ρ_{2} agree on the surface subgroup π_{1}(S). Once we find such structures, the representations ρ_{1}, ρ_{2} induce a homomorphism ρ : π_{1}(M) → P SL(2, ℂ). To conclude that ρ is a holonomy of a hyperbolic structure on M one has to prove that ρ satisfies some further conditions (“Maskit Combination Theorem”; for instance, trivial representations would not work), but at this stage we concentrate only on the existence problem for ρ. We have a pair of representation varieties

V_{i} = R(π_{1}(M_{i}), P SL(2, ℂ)) = Hom(π_{1}(M_{i}), P SL(2, ℂ))//P SL(2, ℂ)

and restriction maps r_{i} : V_{i} → R(π_{1}(S), P SL(2, ℂ)) = V. Our goal is to find a point of intersection between r_{1}(V_{1}) and r_{2}(V_{2}). The key is that the Calabi-Mostow Rigidity Theorem in the dimension 3 just “barely holds”: the dimensions of r_{i}(V_{i}) are 1/2 of the dimension of V; if intersection corresponding to a hyperbolic metric on M exists, then this intersection is transversal (Calabi [33]) and unique (Mostow [154]). If the intersection does not exist, then there is something nongeneric about the varieties r_{i}(V_{i}) (compare this with the fact that middle-dimensional complex affine subspaces in ℂ^{2m} “usually intersect”): the gluing map between M_{1}, M_{2} produces an incompressible torus in M. Thurston reduces the problem of existence of intersection between r_{1}(V_{1}), r_{2}(V_{2}) to a certain “fixed-point problem” and then proves existence of a fixed point.

These Notes contain essentially no new results, I preferred to use whatever I could find in the existing literature. I give sketches of proofs (or just references) mostly in the cases when I was comfortable with the proofs that are already published. I tried to give proofs if they seemed to me more transparent than those in standard references (like the proofs of Mostow Rigidity, Theorem of Brooks, etc.), if they were not published, if they are not very long but are central for our discussion (like Covering Theorem 13.19 and Thurston’s Theorem 3.1, etc.). There are only three theorems in these Notes that seem to be relatively new: I included them since the proofs are not difficult and they provide good illustration of the subject. They are:

Theorem 9.29 on smoothness of the representation varieties for finitely generated Kleinian groups. Theorem 19.16, which states that every finitely generated Kleinian group in P SL(2, ℂ) is isomorphic to a geometrically finite group (the last is more or less straightforward application of Thurston’s Hyperbolization Theorem, the only problem is the presence of torsion). And, finally, an analogue of the Johannson-Waldhausen homeomorphism theorem for a class of 3-dimensional orbifolds, Theorem 7.29. I could not find any reasonably general homeomorphism theorem for orbifolds in the existing literature, there are several attempts to prove such, however they fall short of what we need for the Hyperbolization Theorem. It seems that the basic difficulty is absence of an appropriate generalization of the Haken hierarchy to the category of orbifolds [49].

Otherwise, most of the work that was done in these Notes is assembling the results that are already known in one or another form. There are several deep theorems about 3-manifolds and discrete groups that we will use but their proofs are complicated enough to force me to omit even sketches of the proofs: Haken hierarchy theorem, Waldhausen’s homeomorphism theorem for 3-manifolds Ahlfors finiteness theorem and its companions, Sullivan Rigidity Theorem and Bonahon’s theorem about ends of hyperbolic manifolds.

1Conjecturally such covering always exists.

• Chapter 2: 3-dimensional Topology. We discuss here some basic facts about 3-dimensional manifolds, most of them will be used later in the Notes. There are several good books on this subject, so I skipped most of the proofs.

• Chapter 3: Thurston’s Norm. Our discussion here mainly follows Thurston’s paper [193].

• Chapter 4: Geometry of the Hyperbolic Space. This is probably the most eclectic Chapter of the Notes. I include here some basic material on geometry of spaces of nonpositive and negative curvature as well as of spaces of constant negative curvature, again there is a lot of books on this subject. I did not give much proofs except in cases when proofs are very simple.

• Chapter 5: Teichmüller Theory of Riemann Surfaces. Since there are several excellent books about the Teichmüller Theory, I had tried to give only a bare minimum of material: facts and definitions that will be used in the Notes.

• Chapter 6: Kleinian Groups. This Chapter contains mostly “pre-Thurston” material: results which in some form were known before Thurston came to the field. There are several extremely important “pre-Thurston” theorems on structure of Kleinian groups: Ahlfors finiteness theorem and its companions, Combination Theorems, Kazhdan-Margulis-Zassenhaus lemma and various characterizations of geometrically finite Kleinian groups.

• Chapter 7: Introduction to the Orbifold Theory. We start with basic definitions and examples of orbifolds. Then we introduce a special class of 3-dimensional orbifolds, which I call the class R: they are “locally reflective orbifolds” with some extra restrictions. Basically this is a generalization of the class of atoroidal Haken manifolds whose boundary has zero Euler characteristic. For this class of orbifolds we prove that an isomorphism of fundamental groups is always induced by a diffeomorphism.

• Chapter 8: Complex Projective Structures. I must confess, I had included this Chapter mostly because I like the subject, however we will use the Holonomy Theorem as a technical tool in the discussion of strong convergence.

• Chapter 9: Sociology of Kleinian Groups. This is a Chapter about “collective behavior” of Kleinian groups in families (deformations, algebraic and geometric convergence) and pairs (rigidity theorems and realization of isomorphisms by quasiconformal mappings). Results on algebraic, geometric and strong convergence of sequences of Kleinian groups deal with various aspects of what happens to algebraic structure and geometry of Kleinian groups when we pass to a limit. We prove here the Mostow Rigidity Theorem, so that the proof provides us with some extra information on non-differentiability of quasiconformal mappings conjugating Kleinian groups. It is used in proving the Hyperbolization Theorem.

• Chapter 10: Singular Hyperbolic Metrics and Pleated Surfaces. Here we start descending into the area, which was essentially created by Thurston for purposes of proving the Hyperbolization Theorem. Instead of “pleated surfaces” , which were introduced by Thurston, we will use their combinatorial counterpart: “singular pleated surfaces” which are much easier to handle (I believe that they were invented by Bonahon). The central result of this Chapter is the Covering Theorem 13.19.

• Chapter 11: Ultralimits of Metric Spaces. This Chapter was almost entirely taken from my joint paper with Bernhard Leeb [109]. Ultralimits were introduced into the field by two logicians: Van den Dries and Wilkie, who used them to give another version of Gromov’s proof of his theorem on groups of polynomial growth [204]. Ultralimits provide a convenient formalism for construction of group actions on trees out of divergent sequences of group representations into rank 1 Lie groups. To define ultralimits one needs the Axiom of Choice. If you do not believe in this axiom, you can use for instance [151], [19] or [164] as an alternative.

• Chapter 12: Actions of Groups on Trees. Here we see how to use actions of groups on trees to compactify representation varieties. I also give proofs of some elementary facts about group actions on trees.

• Chapter 13: Measured Laminations, Foliations and Actions of Surface Groups on Trees. Most of the material in this Chapter was introduced by Thurston (part of it was implicit in early works of Dehn and Nielsen). This material is important for understanding Thurston-Otal approach to the hyperbolization of manifolds fibered over circles, however if you are not interested in it you can skip most of the Chapter 13.

• Chapter 14: Rips Theory. The deepest part of the original Thurston’s proof of the Hyperbolization Theorem was certain compactness theorem for sequences of representations. The most efficient (at the present time) way to prove this compactness result is via the “Rips Theory”. Rips’ original proof [171] is still in preparation (hopefully it will not share the fate of Thurston’s preprints). We will use a version of it due to Bestvina and Feighn [21] to derive the needed theorem about fixed points of group actions on trees. An early version of this theorem was proven by Morgan and Shalen [152], it also suffices for our purposes. However the volume of their papers is enormous and the proof is not as transparent as Rips’.

• Chapter 15: Outline of the Proof of the Hyperbolization Theorem. This and the following two Chapters mainly follow Morgan’s paper [150], I used results about group actions on trees as a substitute of certain compactness results of Thurston. We break the proof in several steps and in the following Chapters fill in the details. There are two different cases: the “generic” Case (A) and the “exceptional” Case (B) of manifolds fibered over circles. The Case (A) is discussed in the Chapters 16 and 17. The Case (B) is discussed in the Chapter 18.

• Chapter 16: Reduction of the Generic Case of the Hyperbolization Theorem to the Bounded Image Theorem. This is a slight modification of a portion of Morgan’s paper [150]. I had changed some details which I did not like or did not understand. The existence theorem for hyperbolic metric is reduced to a certain Fixed-Point Theorem (similarly to the fact that the fixed-point set of a map σ : T → T is the intersection of the graph of σ with the diagonal). Standard proofs of such Fixed-Point Theorems require strict contraction property for σ. In our case strict contraction fails, however σ is a contraction and any sequence of iterations σ^{j}(x) is bounded in T. The space T under consideration is a certain Teichmüller space.

• Chapter 17: Bounded Image Theorem. This is the central part of the proof of the Hyperbolization Theorem (the sequence of iterations σ^{j}(x) is relatively compact). I was following here an outline of the proof given in the Morgan’s paper [150].

• Chapter 18: Hyperbolization of Manifolds Fibered Over 𝕊^{1}. The situation with the manifold M fibered over 𝕊^{1} with the typical fiber S is quite different from the “generic” case. In this case there is no fixed point in the space T itself (and σ is an isometry), however there is a fixed point on a “boundary” of T, which gives a hyperbolic structure to the given 3-manifold M. This boundary fixed point [ρ_{∞}] corresponds to a representation ρ_{∞} : π_{1}(S) → Isom(ℍ^{3}). The most difficult part of the proof is to establish existence of [ρ_{∞}], the less difficult part is to understand geometric properties of the group ρ_{∞}(π_{1}(S)). Our discussion essentially follows [162].

• Chapter 19: Hyperbolization of Orbifolds. I explain how to glue orbifolds of the class R from orbifolds of “finite type”. The latter are similar to the class of acylindrical, atoroidal Haken 3-manifolds. Given a geometrically finite hyperbolic structure on orbifolds of finite type we will construct hyperbolic structures of finite volume on orbifolds of the class R. The central part of this Chapter is a Theorem of Brooks: any geometrically finite Kleinian group admits a small deformation which is a subgroup in a lattice. A version of this theorem is used to prove existence of geometrically finite hyperbolic structures on orbifolds of “finite type”. This theorem is used instead of a generalized Andreev’s theorem.

1.3 Beyond the Hyperbolization Theorem

Some hypothesis of the “Big Monster” are necessary for hyperbolicity of M, some are not. The conjectural picture is the following:

Conjecture 1.1 (Thurston’s Hyperbolization Conjecture). Suppose that M is a closed irreducible 3-manifold such that π_{1}(M) is infinite and M is atoroidal. Then M is admits a hyperbolic metric.

Conjecture 1.2 (Thurston’s Geometrization Conjecture). Let M be any closed 3-manifold which does not contain 2-sided projective planes. Then M admits a connected sum decomposition and a decomposition along disjoint incompressible tori and Klein bottles into pieces M_{j} such that each M_{j} is a “geometric manifold”, i.e. admits a complete locally homogeneous Riemannian metric of finite volume. In the case when M is nonorientable and contains 2-sided projective planes the formulation of this conjecture is more complicated, see [180] for details.

In particular this conjecture would imply that homotopy equivalence of closed aspherical 3-manifolds is induced by a homeomorphism. This conjecture includes the Poincare Conjecture and generalized Smith’s Conjecture as particular cases:

Conjecture 1.3 (3-dimensional Poincare Conjecture.) Any 3-manifold which is homotopy equivalent to the 3-sphere is homeomorphic to 𝕊^{3}.

Conjecture 1.4 (Generalized Smith’s Conjecture.) Let Γ be a finite group acting freely on the 3-sphere. Then this action preserves a metric of constant positive curvature on 𝕊^{3}.

It is clear that conjectures 1.1, 1.3 and 1.4 imply the conjecture 1.2.

Conjecture 1.5 (Thurston’s Realization Conjecture). Let M be a geometric 3-manifold and F be any finite group of diffeomorphisms of M. Then there is a geometric structure on M where F acts by isometries.

Thurston claims to have a proof this conjecture in the case when the dimension of the fixed-point set of some nontrivial element of F is at least 1, he also claims the following. Suppose that M is a closed irreducible 3-manifold which admits a finite group of diffeomorphisms such that the dimension of the fixed-point set of some nontrivial element is at least 1, then the Conjecture 1.2 is valid for M. There several papers proving particular cases of Thurston’s claim [92], [118], [212, 213], [23], [40], however Thurston’s claim remains unproven. More general conjecture is

Conjecture 1.6 Let O be a good closed 3-dimensional orbifold. Then O admits decomposition along incompressible spherical and Euclidean suborbifolds into suborbifolds O_{j} so that the following holds. If we attach 3-dimensonal orbifolds covered by the 3-ball to each boundary component of O_{j} which is a spherical 2-orbifold, then each resuling orbifold Ô_{j} admits a geometric structure modelled on one of the eight 3-dimensional geometries.

Several other relevant conjectures:

Conjecture 1.7 Let M be a closed 3-manifold whose fundamental group is not almost Abelian. Then M has finite coverings with arbitrarily large 1-st Betti numbers.

Conjecture 1.8 (W. Thurston) Any hyperbolic 3-manifold of finite volume is finitely covered by a surface bundle over 𝕊^{1}.

Remark 1.9 This is the most doubtful of all Thurston’s conjectures.

Conjecture 1.10 (Weak Hyperbolization Conjecture) Suppose that M is a closed aspherical 3-manifold. Then either π_{1}(M) contains ℤ × ℤ (i.e. M is toroidal) or π_{1}(M) is word hyperbolic.

It is known that non-hyperbolicity of π_{1}(M) implies existence of a “least area” map from a certain compact measured surface lamination of zero Euler characteristic into M (see [153], [68]). One way to prove the Conjecture 1.10 is to show that existence of such map implies that π_{1}(M) contains ℤ × ℤ. Along these lines it was proven in [105] that if M is a closed aspherical Riemannian manifold whose universal cover contains totally geodesic Euclidean plane, then ℤ^{2} ⊂ π_{1}(M). Gabai and Kazez [68, 69] recently proved the Conjecture 1.10 for 3-manifolds which contain “genuine essential laminations”.

Suppose that M is a closed aspherical 3-manifold whose fundamental group G is word hyperbolic. In this case it is known that the ideal boundary ∂G is homeomorphic to the 2-dimensional sphere (see [22]).

Conjecture 1.11 (Cannon’s Conjecture) Under the abobe conditions the action of G = π_{1}(M) on 𝕊^{2} is topologically conjugate to action of the group G′ that consists of holomorphic and antiholomorphic transformations of the standard conformal structure on 𝕊^{2}.

Suppose for a moment that conjectures 1.10 and 1.11 are valid. Then each closed irreducible aspherical 3-manifold M is either toroidal or its fundamental group acts on 𝕊^{2} as a discrete group of conformal and anticonformal transformations. In the former case M is either Haken or Seifert (it follows from the results [144], [66], [37], [179] and Thurston’s geometrization theorem for Haken manifolds). In the latter case G is isomorphic to a discrete subgroup G′ of Isom(ℍ^{3}), i.e. M is homotopy-equivalent to a closed hyperbolic 3-manifold N = ℍ^{3}/G′.

Theorem 1.12 (D. Gabai [67]) Any closed irreducible 3-manifold N which is homotopyequivalent to a hyperbolic manifold, is hyperbolic itself.

Thus in the both “toroidal” and “word hyperbolic” cases of the Conjecture 1.10, the manifold M satisfies Thurston’s hyperbolization conjecture.

2 3-dimensional Topology

2.1 Basics of basics

Notation. Let I denote the compact interval [0, 1]. A compact surface Σ is called pants (or a pair of pants) if it is homeomorphic to the sphere with three holes.

Suppose that S is a closed oriented surface of genus g. Remove from S interiors of disjoint closed disks D_{1}, ..., D_{n}. We say that g is the genus of the resulting surface. A solid torus is the product D^{2} × 𝕊^{1}, where D^{2} is the closed 2-disk.

The handlebody of the genus g is the 3-manifold which is obtained by attaching g solid tori along disjoint boundary disks to the 3-ball. Each handlebody is bounded by the surface of genus g.

By a closed manifold we shall mean a compact manifold without boundary. All manifolds, maps, etc., will be considered either in P L or in the smooth category. This is not a very strong restriction because of

Theorem 2.1 (E. Moise, see [149]) Each topological 3-manifold admits a smooth (and P L) structure. This structure is unique.

However, this choice will allow us to avoid consideration of “wild” embeddings of 1 and 2-dimensional manifolds into 3-manifolds.

Suppose that M is a PL 3-manifold, S, F are PL 2-submanifolds in M. Their intersection is called transversal if it locally looks like:

• either the intersection of two distinct coordinates planes in 3-space;

• or (if say F is a surface with boundary) like the intersection of the coordinate xy-plane in ℝ^{3} with the coordinate yz-half-plane.

An embedding ι : S → M of a surface into 3-manifold is called proper, if ι(∂S) ⊂ ∂M and ι(S) is transversal to ∂M.

A 3-manifold M is called prime if it is not a nontrivial connected sum.

Theorem 2.2 (H. Kneser, J. Milnor, see [89]) Each compact 3-manifold M admits a connected–sum decomposition into prime components. This decomposition is unique under the assumption that M is orientable. (Otherwise we have to assume that there are no 𝕊^{2}×𝕊^{1} factors in the decomposition.)

Remark 2.3 If N is nonorientable then N#ℝP^{3} ≅ N#(𝕊^{2} × 𝕊^{1}).

Theorem 2.4 (The “Dehn Lemma”, see [89].) Suppose that M is a 3-manifold and f : D^{2} → M is such that f is an embedding at some neighborhood A of ∂D^{2} and f^{−1}f(A) = A. Then f|_{∂D^{2}} extends to an embedding g : D^{2} → M.

The usual application of Dehn Lemma is the following: suppose that γ is a simple loop in ∂M which is homotopically trivial in M. Then γ bounds an embedded disk in M.

Theorem 2.5 (The Loop Theorem, see [89].) Let M be a 3-manifold, F be a connected 2-submanifold of ∂M. If π_{1}(F) → π_{1}(M) is not injective then there is a proper embedding

g : (D^{2}, ∂D^{2}) → (M, F)  (1)

such that g(∂D^{2}) is homotopically nontrivial on F.

Theorem 2.6 (The Sphere Theorem, see [89].) Let M be a orientable 3-manifold such that π_{2}(M) ≠ 0. Then there is an embedding g : 𝕊^{2} → M which represents a nontrivial element of π_{2}(M).

Note that this theorem is wrong without the assumption that M is orientable.

A 3-manifold M is called irreducible if it is prime, contains no 2-sided projective planes and is different from 𝕊^{1} × 𝕊^{2}. Considering irreducible manifolds will allow us to avoid the Poincare Conjecture.

Assumption. From now on we shall deal only with connected irreducible manifolds.

Theorem 2.7 (The Annulus Theorem, see [179].) Suppose that M is an orientable 3manifold with boundary, γ_{1}, γ_{2} are two homotopically nontrivial (in M) loops on ∂M which are homotopic in M but not in ∂M. Then there is an embedded annulus (A, ∂A) ⊂ (M, ∂M) so that boundary components of A are not isotopic on ∂M.

Suppose that S is a compact surface. A continuous map f : (S, ∂S) → (M, ∂M) is called essential if:

• it can not be homotoped (relatively to ∂S) to a map f′ : S → ∂M;

• the induced homomorphism π_{1}(f) : π_{1}(S) → π_{1}(M) is injective.

A properly embedded annulus ι : A ↪ M is called an essential annulus if ι is an essential map. Similarly we define essential Moebius bands in M.

Exercise 2.8 Consider the manifold N = T^{2} × [−1, 1], let τ : T^{2} → T^{2} is an involution so that T^{2}/τ is the Klein bottle. Extend τ to τ̂ : (x, t) → (τ(x), −t). Take the manifold M := N/τ̂, it has only one boundary component (the torus T^{2}). The manifold M contains essential annuli as well as essential Moebius bands. Find all of them up to isotopy.

2.2 Incompressible surfaces

At the “first approximation”, incompressible surfaces are those whose embeddings in M^{3} induce monomorphisms of the fundamental groups. Below we give a complete definition.

Let (F, ∂F) ⊂ (M, ∂M) be a surface which is either properly embedded or is contained in ∂M. A compressing disk for F is an embedding (D^{2}, ∂D^{2}) → (M, F) such that: F ∩D^{2} = ∂D^{2} is a (homotopically) nontrivial loop on F (Figure 1).

A boundary compressing disk for F is a triple (D^{2}, A, B) ⊂ (M, ∂M, F) such that:

(i) A ∪ B = ∂D^{2}; (ii) D^{2} ∩ F = B (iii) D^{2} is not isotopic (rel ∂D^{2}) to an embedding which lies in ∂M ∪ F and that meets

each ∂M, F in a disk.

Consider a surface (F, ∂F) ⊂ (M, ∂M) which is either properly embedded or is contained in the boundary of M.

Definition 2.9 A surface (F, ∂F) ⊂ (M, ∂M) as above is called an incompressible surface if it satisfies:

1. F is 2-sided;

2. (i) Either F = D^{2} and ∂F is a homotopically nontrivial loop on ∂M,

(ii) or F is non-simply connected and has no compressing disks and boundary compressing disks.

Figure 1: Compressing disk and boundary compressing disk.

Note that F is not required to be connected.

Exercise 2.10 Let K be a nontrivial knot in 𝕊^{3} and F is the boundary of its regular neighborhood . Then F is incompressible in 𝕊^{3} − K. If F is any connected 2-sided incompressible surface in M^{3} then π_{1}(F) is embedded in π_{1}(M).

To prove Thurston’s Hyperbolization Theorem we shall need a somewhat stronger definition. A compressing annulus for (F, ∂F) ⊂ (M, ∂M) is an embedded annulus

i : (A = I × 𝕊^{1}, {0} × 𝕊^{1}, {1} × 𝕊^{1}) ↪ (M, F, ∂M)  (2)

such that:

(i) i_{∗} : π_{1}(A) → π_{1}(M) is an embedding;

(ii) A ∩ F = {0} × 𝕊^{1} = γ;

(iii) γ is not isotopic to ∂F in F.

Example 2.11 Take M = T^{2} × I. Let F = T^{2} × {0.5}. Then F admits a compressing annulus.

Definition 2.12 (Superincompressible surface.) Suppose that M is a 3-manifold, F ⊂ M is a properly embedded surface. We say that F is superincompressible if it is incompressible and has no compressing annuli.

Example 2.13 Any incompressible surface in a closed 3-manifold is superincompressible.

Definition 2.14 A compact irreducible manifold is called Haken if it contains a superincompressible surface.

Warning 2.15 Lens spaces contain 1-sided surfaces which satisfy the property (2) in the Definition 2.9, but they fail to be incompressible since they are 1-sided. Moreover, lens spaces are not Haken! See [89].

Definition 2.16 (Haken hierarchy.) Start with a connected surface S_{1} in M = M_{1} which is superincompressible and has the largest possible Euler characteristic. Split M along S_{1}. As the result we obtain a manifold M_{2} each of whose components is either Haken or a 3-ball. Then in each Haken component of M_{2} again take a connected surface S_{2} which is superincompressible and has the largest possible Euler characteristic. Continue this process until all components of M_{h} are homeomorphic to the 3-ball.

Theorem 2.17 (F. Waldhausen, W. Haken). Each Haken manifold has a finite Haken hierarchy: a collection of manifolds M_{1}, M_{2}, ..., M_{h} (obtained in the process of decomposition above) together with the gluing data.

For proof see [89], [97].

Maximality of the Euler characteristic of S_{j} is needed to guarantee that the process of decomposition will eventually terminate. Otherwise, say on one of the steps, one can choose a superincompressible surface of non-maximal Euler characteristic.

Example 2.18 (Haken hierarchy.) Start with M which is fibered over a circle. First cut M along the fiber S to obtain M_{2} = S ×[0, 1]. Then cut M_{2} along the “vertical cylinders” of the form γ × [0, 1] (γ ⊂ S) to the get a union of P_{j} × [0, 1], where P_{j} are pairs of pants. Finally, in each P = P_{j} choose two disjoint arcs a, b which connect different boundary components and split P × [0, 1] along a × [0, 1] and b × [0, 1]. The result is the union of balls.

Definition 2.19 Compact manifold M whose boundary is a (possibly empty) collection of surfaces of nonpositive Euler characteristic is called atoroidal if:

(i) each rank 2 Abelian subgroup of π_{1}(M) is conjugate into the image of the fundamental group of some boundary component;

(ii) M is different from an I-bundle over the torus or Klein bottle and from a disc bundle over 𝕊^{1}.

Definition 2.20 A compact manifold M whose boundary is a (possibly empty) collection of surfaces of nonpositive Euler characteristic, is called topologically atoroidal if:

(i) each incompressible torus and Klein bottle in M is isotopic to a boundary component of M;

(ii) M is different from an I-bundle over the torus or Klein bottle and from a disc bundle over 𝕊^{1}.

Example 2.21 Take M = the pair of pants ×𝕊^{1}. Then M is not atoroidal but is topologically atoroidal. Let K be the trefoil knot. Then 𝕊^{3} − Nbd(K) is not atoroidal but is topologically atoroidal. With exception of so called Seifert manifolds that we shall discuss below, being an atoroidal manifold is equivalent to being topologically atoroidal.

Definition 2.22 A compact 3-manifold M is called acylindrical if it does not contain essential annuli. Manifold M is weakly acylindrical if for each properly embedded annulus A ⊂ M the pair of loops ∂A bounds an annulus in ∂M.

Theorem 2.7 implies that M is acylindrical iff the fundamental group π_{1}(M) does not split into a nontrivial amalgamated free product and HNN-extension with amalgamation over ℤ. Note that atoroidal manifold is acylindrical if and only if it is weakly acylindrical.

Recall that a group is called almost abelian (almost nilpotent, etc.) if it contains an abelian (nilpotent, etc.) subgroup of finite index.

Theorem 2.23 If M is compact atoroidal and H = {M_{j}}_{j∈J} is a Haken hierarchy for M, then all manifolds M_{j} are atoroidal or have almost Abelian fundamental groups.

Proof: According to Seifert– Van Kampen Theorem the inclusions M_{i+1} → M_{i} induce monomorphisms of the fundamental groups. Thus, if there is a subgroup A = ℤ^{2} ⊂ π_{1}(M_{i+1}), then A is embedded in π_{1}(M_{i}) as well. We assume that A is not conjugate into fundamental group of any boundary component of M_{i+1}. Suppose that A is conjugate in π_{1}(M_{i}) into the fundamental group of one of the boundary components T_{j} (T_{j} must be a torus). It follows from the algebraic structure of amalgamated free product (or HNN extension) that T_{j} must be a boundary component S_{i} of M_{i+1}. Note that the surface S_{i} is neither a torus nor a Klein bottle. This proves that A is conjugate (in π_{1}(M_{i+1})) into π_{1}(T_{j}). □

Remark 2.24 There is no way to avoid solid tori as members of the Haken hierarchy.

2.3 Existence theorems for superincompressible surfaces

Theorem 2.25 Suppose that M is compact, irreducible and

0 ≠ z ∈ H_{2}(M, ∂M; ℤ)

Then M contains an orientable (possibly disconnected) superincompressible surface S representing the class z.

The proof is based on cut and paste operations. See [97], [89] and Section 3.

Corollary 2.26 Suppose that a 3-manifold M is compact orientable, irreducible, distinct from B^{3} and has nonempty boundary. Then M contains a properly embedded superincompressible surface.

Proof: Note that χ(∂M) ≤ 0, thus H_{1}(M, ℤ) ≠ 0. By duality we conclude that H_{2}(M, ∂M; ℤ) ≠ 0. □

Corollary 2.27 Suppose that 0 ≠ z ∈ H_{2}(M, ∂M; ℤ) is an element which is different from multiple of homology class of any fiber in a fibration of M over 𝕊^{1}. Then any connected component S_{j} of the surface S representing z (as in Theorem 2.25) is not a fiber in a fibration of M over 𝕊^{1}.

Proof: All the components S_{j} are disjoint. Suppose that S_{j} ⊂ S is a fiber in a fibration over 𝕊^{1}. Split M along S_{j}, the result is S_{j} × I which contains the rest of the surface S. Therefore [S] = q[S_{j}], which contradicts our assumptions. □

A stronger existence result is implicit in the paper of Freedman, Hass and Scott [62] and was later explicitly used by Bonahon [24] and in the work of Canary and Minsky [36]. We first formulate the absolute version of this theorem.

Theorem 2.28 Suppose that (M, ds^{2}) is a Riemannian 3-manifold, S ⊂ M is a closed superincompressible surface (which is not necessarily connected), f : S → M is a map homotopic to the inclusion ι : S ↪ M. Then in the 2-neighborhood U of f(S) there is an embedded superincompressible surface Σ which is homotopic to S.

Proof: I first give an informal proof suggested by Bonahon. Recall that the main result of [62] states that a least area map g : S → M homotopic to ι is either

1. an embedding or,

2. a 2-fold covering onto a 1-sided surface.

Note that in the second case we can always approximate g by a homeomorphism g_{ϵ} onto the boundary of a regular neighborhood of g(S). To guarantee existence of a least area map as well as the fact that it lies in U, we change the Riemannian metric on M as follows. Keep the metric in the 0.5-neighborhood V of f(S) as it was before, outside of the 1-neighborhood W of f(S) change the metric ds^{2} to ρ(x)ds^{2}, so that:

• The injectivity radius at each point x ∈ (M − W, ρ · ds^{2}) is at least Area(f(S));

• V ol(B_{x}(1)) ≫ Area(f(S)) for any x ∈ M − V.

Then, consider a sequence of maps f_{n}, n = 1, 2, ... in the homotopy class of f = f_{1}, which is area-minimizing, this sequence must stay inside U. This guarantees both existence of a limiting area-minimizing map f_{∞} and the fact that the image of f_{∞} : S → Σ ⊂ M is contained in U.

However to make these arguments formal one has to overcome some technical problems which do not look simple. For instance, one ingredient of the proof should be a relativized 2-dimensional version of the following theorem due to Gromov [81]:

Theorem 2.29 Suppose that L is a closed aspherical Riemannian n-manifold where the length of the shortest homotopically nontrivial loop is at least ϵ > 0. Then the volume of L is at least ϕ(n, ϵ), where ϕ(n, ϵ) is a universal function which does not depend on L.

Fortunately there is a relatively cheap way around [36]. Namely, Jaco and Rubinstein [98, 99] developed a theory of combinatorial least area surfaces generalizing the classical theory, which is easy to handle. Choose a smooth triangulation T of M. Instead of minimizing area of surfaces in M Jaco and Rubinstein minimize a quantity w(f) for normal simplicial maps f : S → M. All we need to know about the definition of normality is that it includes transversality to the 1-dimensional skeleton T^{(1)} of T. The exact definition of w(f) is not essential here, what is important is that w(f) is at least the number of points in f^{−1}(T^{(1)}). As in [62] it turns out that w-minimizing map g : S → M homotopic to an embedding is an embedding itself. Now, instead of expanding the metric on M outside of W, we just take a sufficiently deep barycentric subdivision of T in M − W and keep the old triangulation in V. The new triangulation T′ of M determines a combinatorial distance between its 3-simplices (distance between any adjacent simplices is equal to 1). For each u ∈ U take a top-dimensional simplex Δ ∈ T′ containing u and let B(u, w(f)) be the union of 3-simplices of T′ within the combinatorial distance at most w(f) from Δ. We require each B(u, w(f)) to be contractible. Now the same arguments as in the case of area-minimizing surfaces imply that image of any w-minimizing π_{1}-injective map g : S → M is contained in W. □

The situation in the case of surfaces S with boundary is slightly more subtle.

Theorem 2.30 Suppose that (M, ds^{2}) is a Riemannian 3-manifold, S ⊂ M is a superincompressible surface, f : (S, ∂S) → (M, ∂M) is a map homotopic to the inclusion ι : (S, ∂S) ↪ (M, ∂M). Assume that f is an embedding near ∂S. Then in the 2neighborhood U of f(S) there is an embedded superincompressible surface Σ which is homotopic to S.

Proof: There is a relative version of the theorem of Freedman, Hass and Scott above (see [62, Section 7] and [98]). Consider the relative minimization problem, i.e. the area-minimization problem (or, in the combinatorial approach, the w-minimization problem) for maps g : S → M which are equal to f on ∂S and homotopic to f relatively to ∂S. Again there exists an area–minimizing surface which is either embedded or is a 2-fold covering of an embedded 1-sided surface. Then we repeat arguments in the proof of the previous theorem. □

2.4 Deforming homotopy-equivalences to homeomorphisms

Theorem 2.31 (F. Waldhausen, see [89]). Suppose that M_{1} is a closed orientable Haken manifold, M_{2} is another irreducible 3-manifold which is homotopy-equivalent to M_{1}. Then any isomorphism of the fundamental groups ϕ : π_{1}(M_{1}) → π_{1}(M_{2}) is induced by a homeomorphism.

Theorem 2.32 (F. Waldhausen, see [89]) Let M_{1} be an orientable Haken manifold, M_{2} be another compact irreducible 3-manifold and

f : (M_{1}, ∂M_{1}) → (M_{2}, ∂M_{2})

be a homotopy-equivalence of pairs. Then f is homotopic to a homeomorphism.

Suppose that M is a compact Haken 3-manifold with boundary. Following [89] we define the peripheral structure of π_{1}(M) as follows. For each boundary component S_{j} ⊂ ∂M (j = 1, ..., k) fix the homomorphism

θ_{M,j} : π_{1}(S_{j}) → π_{1}(M)

induced by the inclusion. Then the collection

P(π_{1}(M)) = {(π_{1}(S_{j}), θ_{M,j}), j = 1, ..., k}

is called the peripheral structure of π_{1}(M).

A subgroup G ⊂ π_{1}(M) is called peripheral if there exists a boundary surface S_{j} so that G is conjugate to a subgroup of θ_{M,j}(π_{1}(S_{j})) in π_{1}(M).

Suppose that M, N are Haken manifolds with the peripheral structures

P(π_{1}(M)) = {θ_{M,j} : π_{1}(S_{j}) → π_{1}(M), j = 1, ..., k};
P(π_{1}(N)) = {θ_{N,j} : π_{1}(T_{j}) → π_{1}(N), j = 1, ..., k}

Define an isomorphism of peripheral structures P(π_{1}(M)) → P(π_{1}(N)) as a collection of isomorphisms ψ : π_{1}(M) → π_{1}(N), ϕ_{j} : π_{1}(S_{j}) → π_{1}(T_{j}) so that

ψ ∘ θ_{M,j} = θ_{N,j} ∘ ϕ_{j}, j = 1, ..., k

Theorem 2.33 (See [89].) Suppose that M, N are Haken manifolds as above, ψ : π_{1}(M) → π_{1}(N) is an isomorphism and

h_{j} : S_{j} → T_{j}, j = 1, ..., k

is a collection of homeomorphisms which induce an isomorphism of the peripheral structures of π_{1}(M) and π_{1}(N). Then this collection of homeomorphisms can be extended to a homeomorphism M → N.

Theorem 2.34 (K. Johannson, see [97], [101].). Suppose that M_{1} is Haken and acylindrical. Then for every irreducible compact 3-manifold M_{2} each homotopy–equivalence h : M_{1} → M_{2} is homotopic to a homeomorphism.

Below we give two simple examples which show that assumptions of Theorem 2.32 and Theorem 2.34 are necessary.

(1) Let P be the pair of pants and T^{∗} be the torus with a hole. Then manifolds P × 𝕊^{1} and T^{∗} × 𝕊^{1} are homotopy-equivalent but not homeomorphic.

(2) Consider four copies S_{1}, S_{2}, S_{3}, S_{4} of the torus with a hole. On the boundary of a solid torus T̂ draw four disjoint parallel loops which are (homotopically) nontrivial in T̂. These loops separate the boundary of T̂ into four annuli A_{j}. Now glue the manifolds N_{j} = S_{j} × I along the boundary annuli ∂S_{j} × I ≅ A_{j} to T̂. The result is a compact Haken 3-manifold M. There is an obvious automorphism φ of the fundamental group π_{1}(M) which preserves π_{1}(N_{1}), π_{1}(N_{3}) but interchanges π_{1}(N_{2}), π_{1}(N_{4}). Clearly φ is not induced by a self-homeomorphism of the manifold M.

2.5 Paired 3-manifolds

A paired manifold is a pair (M, P) such that:

• M is a compact orientable irreducible 3-manifold;

• P ⊂ ∂M is a disjoint union of incompressible tori and annuli P_{j};

• no two components of P are isotopic in ∂M;

• every Abelian noncyclic subgroup of π_{1}(M) is conjugate to π_{1}(P_{j}) for some j;

• there are no essential cylinders

(A, ∂A) ↪ (M, P)

The submanifold P is called the designated parabolic locus of M: its components will correspond to cusps in a complete hyperbolic metric that we will construct on M − P in the process of proving Thurston’s Hyperbolization Theorem. We let ∂_{0}M := ∂M − P. Any paired manifold (M, P) satisfies the following:

Theorem 2.35 Suppose that π_{1}(M) splits into a nontrivial free product A ∗ B, so that the fundamental group of each component P_{j} is conjugate either into A and or into B. Then M contains a compressing disk (D, ∂D) ⊂ (M, ∂_{0}M).

Theorem 2.36 Suppose that π_{1}(M) splits into a nontrivial amalgamated free product A∗_{C} B or HNN extension A∗_{C} with infinite cyclic amalgamated subgroup C, so that the fundamental group of each component P_{j} is conjugate either into A and or into B. Then M contains an essential annulus (or Moebius band) (A, ∂A) ⊂ (M, ∂_{0}M).

We will discuss proofs of these theorems in Section 11.2.

For future purposes I will make the following convention for Haken hierarchies of paired manifolds. If (M, P) is paired, S ⊂ M is a superincompressible surface, then we choose S within its isotopy class so that:

• If a α component of ∂S is isotopic in ∂M to a boundary component of ∂P, then α ⊂ P.

• S minimizes the number of components of ∂S ∩ P in the isotopy class of S.

2.6 Seifert manifolds

This is a very important class of 3-maifolds, so we will give several equivalent definitions.

Definition 2.37 A Seifert manifold is a compact 3-manifold which admits foliation by circles.

Definition 2.38 In the class of compact prime 3-manifolds with infinite fundamental groups, Seifert manifolds M are those whose fundamental groups have infinite normal cyclic subgroups. Their fundamental groups split as:

1 → ℤ → π_{1}(M) → F → 1  (3)

where F is a discrete group of isometries of 𝕊^{2} or 𝔼^{2} or ℍ^{2}.

Definition 2.39 In the class of compact 3-manifolds with infinite fundamental group, Seifert manifolds are those which are finitely covered by circle bundles over surfaces.

Example 2.40 The sphere 𝕊^{3} is a Seifert manifold (𝕊^{1}-fibration is the Hopf fibration.) If S is a compact surface then S × 𝕊^{1} is a Seifert manifold. The total space of the unit tangent bundle of a closed Riemannian surface is a Seifert manifold.

Theorem 2.41 In the class of compact prime 3-manifolds with infinite fundamental groups all three definitions of Seifert manifolds are equivalent.

Proof is a combination of [54], [89], [179], [66], [37].

2.7 Decomposition Theorem

Theorem 2.42 (“Torus theorem”). Suppose that M is an orientable 3-manifold, ℤ^{2} ⊂ π_{1}(M) is a nonperipheral subgroup. Then either M contains an embedded incompressible torus which is not boundary-parallel or M is a Seifert manifold.

This theorem was originally proven for Haken manifolds (see [100]) and later– for all 3manifolds in a sequence of works by Scott [179], Tukia [202], Mess [144], Gabai [66], Casson and Jungreis [37]. Theorem 2.42 implies

Theorem 2.43 (The JSJ Decomposition Theorem, W. Jaco, P. Shalen, K. Johannson). Suppose that M is a Haken 3-manifold whose boundary has zero Euler characteristic. Then there is a collection C of disjoint embedded incompressible tori and Klein bottles in M such that M − C consist only of Seifert manifolds and atoroidal manifolds. Minimal (with respect to the inclusion) collection C_{min} is unique up to isotopy. The decomposition of M into components M − C_{min} is called the Jaco-Shalen-Johannson decomposition of M (the JSJ decomposition).

Now we can formulate the central theorem of this book:

Existence:

Theorem 2.44 (Hyperbolization Theorem, W. Thurston) Suppose that M is a compact Haken atoroidal 3-manifold, whose boundary has zero Euler characteristic. Then the interior of M admits a complete hyperbolic metric of finite volume. By abusing notations we shall say that M is hyperbolic.

More generally we have:

Theorem 2.45 Suppose that (M, P) is a compact atoroidal Haken paired 3-manifold. Then M admits a complete geometrically finite hyperbolic structure with the parabolic locus P. (I will give necessary definitions later.)

Uniqueness:

Theorem 2.46 (D. Mostow [154], Mostow Rigidity Theorem) Suppose that two complete hyperbolic manifolds of finite volume M, N have isomorphic fundamental groups and dimension n > 2. Then any isomorphism of their fundamental groups can be realized by an isometry M → N.

We will give a proof of Mostow Rigidity Theorem in Section 9.4. The proof of Theorem 2.44 that we shall discuss in this book, is based on a wide range of mathematical technique:

(a) discrete groups of isometries of hyperbolic space (Kleinian groups),

(b) 3-dimensional topology,

(c) Teichmüller theory,

(d) pleated surfaces,

(e) actions of groups on trees.

2.8 Characteristic submanifolds in 3-manifolds

In this section we discuss the way how a compact Haken manifold can fail to be acylindrical. Consider a compact 3-manifold pair (N, R), where R ⊂ ∂N is an incompressible surface. There exists a 3-manifold subpair (X, S) ⊂ (N, R), where S ⊂ ∂X, X is possibly disconnected codimension zero submanifold in N, which is called a characteristic subpair of (N, R) and which satisfies the following properties:

1. each component of (X, S) is either a pair (I–bundle, ∂I–subbundle) over a surface or is a solid torus fibered by circles, where S is the union of fibered annuli;

2. ∂X ∩ ∂N = S;

3. components of ∂_{1}(X) = ∂X − S are essential annuli;

4. any essential incompressible annulus (and Moebius band) (A, ∂A) → (N, R) is homotopic (as a map of pairs) into (X, R);

5. the characteristic subpair is unique up to isotopy.

Theory of characteristic subpairs was developed in [100], [101], [97].

Suppose that P = ∂N − R and (N, P) is a paired manifold. Then we call the characteristic subpair (X, S) ⊂ (N, R) to be the characteristic pair of the paired manifold (N, P). The decomposition of (N, P) into the union of components of X and N − int(X) is called the canonical JSJ (Jaco-Shalen-Johannson) decomposition of (N, P). Components N_{j} of N − int(X) are:

• relatively acylindrical in the sense that any essential annulus (and Moebius band) in N_{j} must intersect P ∪ (N_{j} ∩ X);

• relatively incompressible in the sense that any compressing disk for ∂N_{j} must intersect P ∪ (N_{j} ∩ X).

PICTURE!!!!

However these components could be neither acylindrical nor incompressible in the usual sense. Far-reaching generalization of the canonical JSJ decomposition is developed in [172], [173], [51], [52], [63].

Example 2.47

Let N_{j}, j = 1, 2 be a pair of acylindrical manifolds with nonempty boundary surfaces of the genus 2. Choose a pair of simple homotopically nontrivial loops γ_{j} ⊂ ∂N_{j} and a compact oriented surface H with two boundary components. Glue H to N_{1} ∪ N_{2} so that ∂H is glued to the union γ_{1} ∪ γ_{2}. Then take an I-bundle I(H) = I × H over H and use it to “thicken” the surface H in the complex N_{1} ∪ H ∪ N_{2}. The result is a 3-manifold N. Then the pair (I(H), {±1} × H) is the characteristic subpair in (N, ∂N).

Example 2.48

This is example when the characteristic subpair contains a foliated solid torus. Take two copies of the manifold I × F where F is a closed orientable surface of genus 2, choose a separating simple homotopically nontrivial loop γ on F × {1}, let N (γ) be a regular neighborhood of γ on F × {1}. Let V ⊂ U be regular neighborhoods of γ in I × F so that boundaries of U and V are disjoint. Glue two copies of I × F along N (γ). The result is a manifold L which admits a rotation of order 2 around γ so that the corresponding cyclic group ℤ_{2} acts freely on γ. Now take the quotient of L by ℤ_{2}. The result is a manifold N. Characteristic submanifold in N consists of two components which are: a copy of I × F − int(U) and the projection of V to N.

Now we return back to the discussion of characteristic subpairs. Choose a component (X_{j}, S_{j}) of (X, S) which is a solid torus. Then ∂X_{j} − S_{j} is a union of essential annuli in N. Thicken these annuli and consider them as I-bundles. Add to them all components of (X, S) which are I-bundles. The result is a submanifold W ⊂ N (which is an I-bundle). W is called the window in N. The frontier ∂_{1}W = cl(∂W − ∂N) is the union of essential annuli. Then the property (4) of the characteristic subpair implies that each component L of N − intW is

(a) either a solid torus (which appears as above),

(b) or all essential annuli in (L, ∂N ∩ L) are parallel in L to essential annuli on the boundary of the window, i.e. into (∂W, ∂_{1}W ∩ L).

2.9 3-manifolds fibered over 𝕊^{1}

Suppose that (M, P) is a paired orientable 3-manifold, where P = ∂M is a collection of incompressible boundary tori. Let S ⊂ M be a connected superincompressible surface. Define (M′, P′) as a compact paired manifold obtained by splitting M along S. Note that the manifold M′ − P′ has two boundary components which are homeomorphic copies of S. We assume that at least one component M′_{1} of M′ is an interval bundle over a surface Σ_{1} so that P′_{1} := P′ ∩ M_{1} is the interval bundle over ∂S_{1}. We will abbreviate this by saying: (M_{1}, P′_{1}) is an interval bundle over a surface. There are two cases that may occur.

Case 1. M′ is connected. Since ∂M′ − P′ has two components we conclude that M_{1} = M′ is the trivial interval bundle:

M′ = Σ_{1} × I

Thus the manifold M is a surface bundle over 𝕊^{1} with the fiber Σ_{1}.

Case 2. M′ consists of two components M′_{1}, M′_{2}. Each of these manifolds contains exactly one component of ∂M′_{j} − P′. Then the manifold M′_{1} is a nontrivial interval bundle, the base of the bundle has to be non-orientable since the manifold M is orientable.

Subcase 2a. The manifold M′_{2} is also a nontrivial interval bundle over a surface Σ_{2}. Note that the base of this fibration is uniquely determined by the topology of ∂M′_{2} − P′ and hence it is homeomorphic to Σ_{1}, which is a nonorientable surface.

Proposition 2.49 Under the conditions above there exists a 2-fold covering N → M so that:

• N is fibered over 𝕊^{1};

• the lift of Σ_{1} to N is a fiber of this fibration;

• this lift is connected and isotopic to a component of the lift of S.

Proof: Let Σ ≅ Σ_{1} ≅ Σ_{2}, then S → Σ is the 2-fold orientation covering over Σ. Now construct the covering N → M as follows. We assume that Σ_{1}, Σ_{2} are embedded in M as zero sections of the corresponding interval bundles M′_{1}, M′_{2}. The union Σ_{1} ∪ Σ_{2} = Z determines a nontrivial class [Z] in H_{2}(M, ∂M; ℤ/2) (by the Meyer-Vietoris sequence). Note that the homology class of S in H_{2}(M, ∂M; ℤ/2) is trivial. By duality [Z] defines a nontrivial homomorphism H_{1}(M, ℤ/2) → ℤ/2. Thus we get an epimorphism

ψ : π_{1}(M) → H_{1}(M, ℤ/2) → ℤ/2

whose kernel defines the covering N → M. Since π_{1}(S) ⊂ π_{1}(Σ_{1}) lies in the kernel of ψ we conclude that the lift Σ̃_{1} of Σ_{1} to N is a single orientable surface homeomorphic to S. The same is true for Σ_{2}. The complement M − Z is a trivial interval bundle S × I over S, its lift to N consists of two homeomorphic components bounded by Σ̃_{1} ∪ Σ̃_{2}. Thus the whole manifold N is a surface bundle over 𝕊^{1}. □

Definition 2.50 In the circumstances of Proposition 2.49, the surface S ⊂ M is called a virtual fiber in a fibration over 𝕊^{1}.

Subcase 2b. The manifold M′_{2} is not an interval bundle over a surface. In this case as in the proof of Proposition 2.49 the surface Σ_{1} defines a cycle Z in H_{2}(M, P; ℤ/2) which by duality corresponds to an epimorphism

ψ : π_{1}(M) → H_{1}(M, ℤ/2) → ℤ/2

whose kernel defines a covering N → M. The surface Σ_{1} lifts to this covering as an orientable surface F ≅ S. By the assumption the complement N − F is not an interval bundle over S. It proves the following:

Proposition 2.51 Under assumptions of the Subcase 2b, the manifold (M, P) has a 2-fold covering (N, Q) which contains a nonseparating superincompressible surface F so that (N − F, Q − F) is not an interval bundle over a surface.

Let S be a compact surface (may be with boundary). A homeomorphism τ : S → S is called aperiodic if for each nontrivial nonperipheral element γ ∈ π_{1}(S) we have the following:

for every integer n ≠ 0 the elements τ_{∗}^{n}(γ) and γ are not conjugate in π_{1}(S).

Suppose now that S has negative Euler characteristic. The fundamental group of the mapping torus M_{τ} (of τ) splits as

1 → π_{1}(S) → π_{1}(M_{τ}) → ℤ → 1

where a generator t of ℤ acts on π_{1}(S) as τ_{∗} (up to composition with an inner automorphism).

It is an easy exercise to show that τ is aperiodic if and only if the mapping torus M_{τ} is an atoroidal 3-manifold. For instance, if π_{1}(M_{τ}) is not atoroidal, then there is a subgroup ℤ × ℤ in it which does not come from one of the boundary tori. Hence thre has to be an element γ ∈ π_{1}(S) which is not peripheral so that t^{m}γ = γt^{m}, i.e. τ_{∗}^{m}(γ) = γ.

2.10 Baumslag-Solitar relations in 3-manifold groups

Suppose that N is a Haken manifold, Σ′, Σ′′ ⊂ ∂N are disjoint incompressible subsurfaces, γ′ ∈ π_{1}(Σ′) and γ, γ′′ ∈ π_{1}(Σ′′) are nontrivial elements. Let α′ ∈ π_{1}(Σ′), α′′ ∈ π_{1}(Σ′′) denote primitive roots of γ′, γ′′. Assume that γ, γ′, γ′′ are conjugate in π_{1}(N). Let Σ := Σ′ ∪ Σ′′.

Lemma 2.52 Under the above conditions gα′g^{−1} = α′′ and γ, γ′′ are conjugate in π_{1}(Σ′′).

Proof: Take the characteristic submanifold (X, S) ⊂ (N, Σ). Then there exists a component (X_{j}, S_{j}) ⊂ N so that γ, γ′, γ′′ are represented by loops ℓ, ℓ′, ℓ′′ on ∂X_{j} ∩ Σ, the loops ℓ, ℓ′, ℓ′′ are homotopic in X_{j}. There are two cases: either (N_{j}, S_{j}) is an interval bundle over a surface Σ_{j} or X_{j} is a solid torus. In the both cases the assertions of lemma are obvious. □

Suppose that a, b are elements of a group G. These elements satisfy a Baumslag-Solitar relation if

a^{p} = ba^{q}b^{−1} for some p, q ≠ 0  (4)

Note that if p = q, then the relation reduces to the “torus” relation

cb = bc where c := a^{p}

If p = −q then we get the “Klein bottle” relation:

cb = bc^{−1} where c := a^{p}

If |p| ≠ |q| and a ≠ 1, then we say that the Baumslag-Solitar relation (4) is nonelementary.

Theorem 2.53 (W. Jaco, P. Shalen, [100]) Suppose that G is the fundamental group of a Haken 3-manifold M, then there are no nontrivial elements a, b ∈ G which satisfy a nonelementary Baumslag-Solitar relation.

Note that using the construction from the Example 2.48 it is easy to get examples of Haken 3-manifolds whose fundamental groups satisfy elementary Baumslag-Solitar relations with |p| = |q| > 1. In general it is unknown whether elements of fundamental groups of 3-manifolds can satisfy a nonelementary Baumslag-Solitar relations (it would follow from Thurston’s Geometrization Conjecture that they cannot). However if M is an aspherical 3-manifold whose fundamental group is itself a Baumslag-Solitar group:

BS(p, q) := ⟨a, b|a^{p} = ba^{q}b^{−1}⟩

then the Baumslag-Solitar relation must be elementary. The reason is simple: BS(p, q) has infinite abelinization and, thus, M is homotopy-equivalent to a Haken manifold.

Sketch of the proof: I will outline a proof of Theorem 2.53. The proof is done by induction on the length of Haken hierarchy of M. If M is a ball there is nothing to prove. Assume that G = π_{1}(M) does have a nonelementary Baumslag-Solitar relation.

Lemma 2.54 Suppose that G = X ∗_{W} Y (amalgamated free product) or G = X∗_{W} (HN-N extension). Then for every pair of elements a, b ∈ G which satisfy a nonelementary Baumslag-Solitar relation, the element a is conjugate either into X or into Y.

Proof: The best proof of this lemma that I know is via actions of groups on trees. So, skip it or read before the material of Chapter 11. Amalgamated free product as well as decomposition as HNN-extension corresponds to an effective action of G on a simplicial tree T so that the quotient T/G is either a segment [pq] (in the case of amalgamated free product) or is a circle with one vertex p (in the case of HNN-extension). Stabilizers (in G) of the lifts of p, q into T are subgroups conjugate either to X or to Y. Elements of G which are not conjugate into X, Y act freely on T (they are called hyperbolic elements). Hyperbolic elements have the property: each hyperbolic element g has an unique invariant geodesic Ax_{g} ⊂ T, where g acts as a translation. The translational length ℓ_{T} (g) is the distance d(x, gx), x ∈ Ax_{g}. This distance is an invariant of the conjugacy class of g. If the Baumslag-Solitar relation (4) is satisfied, elements a^{p}, ba^{q}b^{−1}, a^{q} have the same translational length. If ℓ_{T} (a) > 0 then |p| = |q|, i.e. the Baumslag-Solitar relation is elementary. Otherwise, a has a fixed point on T, i.e. it is conjugate either into X or into Y. □

So, let S ⊂ M be an incompressible surface properly embedded into M, the manifold N is obtained by splitting M along S. Let S′, S′′ ⊂ ∂N be incompressible surfaces corresponding to S, τ : S′ → S′′ be a gluing map such that N/τ = M. There are two possible cases: either N is connected or it consists of two components. Let me consider the former case (the case of two components is similar), then G is an HNN extension of π_{1}(N):

G = ⟨π_{1}(N), t|tgt^{−1} = τ_{∗}(g), g ∈ π_{1}(S′), tgt^{−1} ∈ π_{1}(S′′)⟩

By the induction hypothesis, π_{1}(N) has no nonelementary Baumslag-Solitar relations. According to Lemma 2.54, the only way G = π_{1}(M) can satisfy a nonelementary Baumslag-Solitar relation is that there are nontrivial elements a^{p}, a^{q} ∈ π_{1}(N) which are conjugate in π_{1}(M). After choosing a, b within their π_{1}(N)-conjugacy classes, we can assume that the relation is:

t^{s}a^{q}t^{−s} = a^{p}

Let me consider the case s = 1. Then the conjugation between the elements a^{p}, a^{q} is given by a map α : A = 𝕊^{1} × I → M so that:

• α_{0} = a^{p}, α_{1} = a^{q};

• α(•, 0.5) : 𝕊^{1} → S

Thus α(•, 0.5) gives us two nontrivial loops γ′, γ′′ on S′, S′′. Now the assertion of Theorem follows from Lemma 2.52. □

3 Thurston’s Norm and Incompressible Surfaces

In this Chapter we prove the following theorem:

Theorem 3.1 (W. Thurston [193]) Suppose that M is a compact atoroidal orientable 3manifold such that rank H_{2}(M, ∂M; ℤ) ≥ 2. Then M contains an embedded superincompressible surface which is not a fiber in a fibration of M over 𝕊^{1} and which represents a nontrivial element of H_{2}(M, ∂M; ℤ).

We give a proof of this theorem in the Section 3.3. Our proof is essentially the same as Thurston’s and we will only clean up some of the details in [193].

3.1 Norms defined over ℤ

Consider the space ℝ^{n} with the lattice ℤ^{n} embedded in the standard way. We suppose that p : ℤ^{n} → ℤ is a non-negative function which satisfies the following axioms:

1. p is linear along rays, i.e. p(m · z) = |m|p(z) for each z ∈ ℤ^{n}, m ∈ ℤ;

2. p is convex over ℤ, i.e. for each k, m ∈ ℤ and z, w ∈ ℤ^{n} we have:

p(k · z + m · w) ≤ |k|p(z) + |m|p(w)

3. p is nondegenerate, i.e. p(z) = 0 iff z = 0.

Function p satisfying these properties is called a norm defined over ℤ.

Theorem 3.2 Each norm defined over ℤ extends to a (usual) norm defined on the whole space ℝ^{n}.

Proof: Suppose that α ∈ ℚ^{n} −{0}. Then there exists m ∈ ℤ such that m·α ∈ ℤ^{n} and we let p(α) = p(m · α)/m. This extension of the function p to ℚ^{n} is well-defined since p is linear on rays in ℤ^{n}. Clearly the function p : ℚ^{n} → ℚ is linear on rays and ℚ-convex in the sense that p(kα + mβ) ≤ |k|p(α) + |m|p(β) for all k, m ∈ ℚ and α, β ∈ ℚ^{n}.

Lemma 3.3 Sublevel sets of the function p : ℚ^{n} → ℚ are bounded.

Proof: Each sublevel set {p(α) ≤ c} is ℚ-convex, thus if this set is unbounded, it must contain a complete ℚ-ray {q · α : q ∈ ℚ_{+}} for some α ≠ 0. However p(kα) = kp(α) → ∞ as k → ∞. Contradiction. □

Therefore p is uniformly continuous and hence admits a continuous extension to a convex function p : ℝ^{n} → ℝ_{+}. To prove that p is a norm we have to check that that p(α) = 0 if and only if α = 0.

Lemma 3.4 p(α) = 0 if and only if α = 0.

Proof: Suppose p(α) = 0 and α ≠ 0. Then p(tα) = 0 for all t ∈ ℝ. Let ϵ = 1/2, then p(z) ≥ 2ϵ for all z ∈ ℤ^{n} − {0}. The sublevel set C(ϵ) = {p(σ) < ϵ : σ ∈ ℝ^{n}} is an open convex cone since it contains the line ℝ · α and p is a continuous convex function. Thus C(ϵ) must contain at least one integral point z ∈ ℤ^{n} − {0}. Contradiction. □

This finishes the proof of Theorem 3.2. □ We retain the name norm defined over ℤ for the extension of p to ℝ^{n}. Let

B^{(p)}(r) = {v ∈ ℝ^{n} : p(v) ≤ r}

denote the ball of radius r centered at zero, with respect to the norm p.

Theorem 3.5 Any norm p : ℝ^{n} → ℝ defined over ℤ has the following property. Suppose that z is an element of ℤ^{n}. Then there exists a linear function l such that:

• l(z/p(z)) = 1;

• l : ℤ^{n} → ℤ, i.e. the function l is defined over ℤ;

• the operator norm of l with respect to p is equal to 1, i.e. |l(α)| ≤ 1 for all α ∈ B^{(p)}(1).

Proof: We consider the sequence of “integral balls” of the radius r in ℝ^{n}: B^{(p)}(r) ∩ ℤ^{n}, where r ∈ ℤ_{+} · p(z) . Then

⋃_{r∈ℤ^{+}} \frac{1}{r} · (B^{(p)}(r) ∩ ℤ^{n}) = B^{(p)}(1) ∩ ℚ^{n}

Note that the integral vector \frac{r}{p(z)} · z belongs to the boundary of B^{(p)}(r) for each r as above. The convex hulls C(r) of B^{(p)}(r) ∩ ℤ^{n} are finite polyhedrons whose faces are defined over ℤ in the sense that each top-dimensional face is contained in a zero level set of a linear function with integer coefficients. Thus for each point \frac{r}{p(z)} · z there is linear function l_{r} with integer coefficients such that l_{r}(\frac{r}{p(z)} · z) = r and the level set \{l_{r}(w) = r\} is disjoint from the interior of C(r). Hence l_{r}(z/p(z)) = 1 and the level set {l_{r}(v) = 1} is disjoint from the interior of \frac{1}{r}C(r). However the convex sets \frac{1}{r}C(r) exhaust the unit ball B^{(p)}(1). Thus operator norms (with respect to p) of the linear functions l_{r} are convergent to 1 as r → ∞. This means that the sequence of linear functions l_{r} with integer coefficients is subconvergent as r → ∞ to a linear function l whose norm is equal to 1. Clearly l has integer coefficients and l(z/p(z)) = 1. □

The ratios z/p(z), z ∈ ℤ^{n}, are dense on the unit sphere ∂B^{(p)}(1). Thus the unit ball B_{x}(1) is the intersection of the sublevel sets {l ≤ 1}, where linear functions l are as in Theorem 3.5. However the number of such linear functions l is finite. We conclude that B_{x}(1) is a polyhedron with finitely many faces which are given by integral linear equations.

Corollary 3.6 Suppose that v is a vertex of the polyhedron B^{(p)}(1). Then there exists an element z ∈ ℤ^{n} such that v = z/p(z).

3.2 Variation of fiber-bundle structure

Suppose that M is a compact 3-manifold fibered over 𝕊^{1}, we use the nottaion F_{t} (t ∈ 𝕊^{1}) for the fibers of this fibration F. Fibers F_{t} are transversal to the boundary of M, we assume that M is provided with a Riemannian metric such that F_{t} are orthogonal to ∂M. Tangent planes to the fibers F_{t} define a plane subbundle P ⊂ T (M). Let L denote the unit vectorfield on M which is orthogonal to P. Along each boundary curve ∂F_{t} we choose the unit tangent field X on F_{t} which is orthogonal to ∂M. Thus we get a section X of P |_{∂M}. The pair (P, X) determines an element of H_{2}(M, ∂M)^{∗} = H^{1}(M) as follows. Suppose that h : (Σ, ∂Σ) → (M, ∂M) is a proper smooth map from a surface Σ which is a representative of a relative cycle ζ ∈ Z_{2}(M, ∂M). Then h^{∗}(P) is a 2-dimensional vector bundle over Σ with a prescribed section h^{∗}(X) over ∂Σ. There is a well-defined obstruction (relative Euler number) e(h^{∗}(P), X)) ∈ ℤ to the extension of h^{∗}(X) to a nonzero section of the bundle h^{∗}(P). Thus we define τ ∈ H_{2}(M, ∂M)^{∗} as τ([ζ]) = e(h^{∗}(P), X). It is easy to see that τ is well-defined. Obviously for the relative class [ζ] ∈ H_{2}(M, ∂M) which is represented by a fiber F_{t} we get: τ([ζ]) = χ(F_{t}).

There is a closed nondegenerate integral 1-form θ on M whose kernel is the subbundle F. Namely, if f : M → 𝕊^{1} is the fibration, then θ is the pull-back under f of the angle form dt from 𝕊^{1}. The converse to this is true as well:

Theorem 3.7 (D. Tishler, [198]) Suppose that ω is a closed nondegenerate 1-form on M which has integral periods (i.e. it determines an element of H^{1}(M, ℤ)). Assume that the restriction of ω to ∂M is nondegenerate. Then there exists a fibration G of M over 𝕊^{1} such that fibers of G are tangent to the kernel-distribution of ω.

Sketch of the proof: Choose a base-point p ∈ M and consider indefinite integrals

f(q) = ∫_{p}^{q} ω

The function f : q ↦ ℝ/ℤ is well-defined, smooth and local calculation shows that f has maximal rank at each point. Thus f is a fibration. □

Now suppose that ω is a closed rational nondegenerate 1-form on M which is sufficiently close to θ, namely we assume that the kernel-distribution of ω is transversal to the vectorfield L. Since ω is rational, the kernel-distribution Q of ω is tangent to a fibration G of M by surfaces G_{t}, t ∈ 𝕊^{1}.

Fibers of G determine a relative class [ξ] ∈ H_{2}(M, ∂M).

Lemma 3.8 τ([ξ]) = χ(G_{t}).

Proof: Since both F and G are transversal to the vectorfield L, we conclude that for each fiber G = G_{t} of G:

P |_{G} ≅ T (M)|_{G}/Span(L) ≅ Q|_{G}

Thus the bundles P |_{G} and Q|_{G} are isomorphic and moreover this isomorphism λ carries the section X (of P |_{∂G}) to a tangent vectorfield λ(X) on G which is normal to ∂G. Therefore the obstruction χ(G) to the extension of λ(X) to a nonzero field on G is the same as the relative Euler number τ([ξ]) = e(P |_{G}, X). □

3.3 Thurston’s norm

We consider a compact irreducible atoroidal orientable manifold M with (possibly empty) incompressible boundary of zero Euler characteristic. Suppose that ξ ∈ H_{2}(M, ∂M; ℤ) is a relative homology class. Define the Thurston’s norm x(ξ) as

x(ξ) = ∥ξ∥ := min{|χ(S)| : (S, ∂S) ⊂ (M, ∂M) is an embedded surface

representing the class ξ}

Since the manifold M is atoroidal, x(ξ) ≠ 0 for each ξ ≠ 0. There is a generalization of Thurston’s norm which is defined by taking minimum over immersed surfaces representing the given homology class. It turns out that two norms coincide, see [65], [167].

Lemma 3.9 Every element ξ ∈ H_{2}(M, ∂M; ℤ) is represented by an embedded oriented surface S. If ξ/n ∈ H_{2}(M, ∂M; ℤ) for some n > 0, then any surface S representing ξ is a union of n disjoint subsurfaces S_{j} each representing ξ/n.

Remark 3.10 The surfaces S_{j} need not to be connected.

Proof: Recall that (by duality) each element ζ ∈ H_{2}(M, ∂M; ℤ) determines an element ζ^{∗} ∈ H^{1}(M; ℤ) ≅ H^{1}(π_{1}(M), ℤ), the last is a homomorphism ζ^{∗} : π_{1}(M) → ℤ. The homomorphism ζ^{∗} is induced by a PL map f : M → 𝕊^{1}; if c is a regular value of f then the surface F = f^{−1}(c) is embedded, has canonical orientation and represent the class ζ. This proves the first assertion of Lemma (cf. Lemma 2.25).

Suppose that ζ ∈ H_{2}(M, ∂M; ℤ) is represented by an embedded relative 2-cycle F. The image of the 1-st homology group of F in H_{1}(M, ℤ) lies in the kernel of ζ^{∗}, hence the restriction of f to F is homotopic to a constant and we can choose f within its homotopy class to be constant on F. We can also assume that f has regular value at c = f(F). The inverse image f^{−1}(c) can be larger then the surface F. However, since f^{−1}(c) is homologous to F, we conclude that F_{0} = f^{−1}(c) − F is homologically trivial.

If two maps f, g : M → 𝕊^{1} correspond to the same homology class ζ, then f is homotopic to g (since they induce the same homomorphism of π_{1}(M)). Now let’s use this to prove the second assertion of Lemma. Suppose that α = ξ/n ∈ H_{2}(M, ∂M; ℤ), f_{α}, f_{ξ} : M → 𝕊^{1} are the corresponding maps and the subsurface S ⊂ f_{ξ}^{−1}(c) represents the homology class ξ. Let λ : 𝕊^{1} → 𝕊^{1} be the n-fold covering, the inverse image λ^{−1}(c) is a set {b_{1}, ..., b_{n}}. Hence the function λ ∘ f_{α} corresponds to the class nα = ξ; this function is homotopic to f_{ξ}. Thus f_{ξ} lifts to a function f̃_{ξ} : M → 𝕊^{1} via the covering λ. Therefore the surface

S ∪ S_{0} = f_{ξ}^{−1}(c) = ⋃_{i=1}^{n} f̃_{ξ}^{−1}(b_{i})

is a disjoint union of n subsurfaces S_{1}, ..., S_{n}. Recall that the relative homology class [S_{0}] is trivial. Since the function f̃_{ξ} is homotopic to f_{α}, we conclude that each of these subsurfaces represents the homology class α. □

From now on we shall assume that all components of surfaces representing homology classes in H_{2}(M, ∂M; ℤ) have negative Euler characteristic. (We can make this assumption since ∂M is incompressible and M is atoroidal and aspherical.)

Theorem 3.11 Suppose that M is a 3-manifold as above. Then the function

x : H_{2}(M, ∂M; ℤ) → ℤ

is a norm defined over ℤ.

Proof: According to Lemma 3.9 we have:

x(nξ) ≥ |n|x(ξ) for all n ∈ ℤ

The opposite inequality is obvious. Thus x(nξ) = |n|x(ξ) for all n ∈ ℤ: x is linear on rays.

Lemma 3.12 If α, β ∈ H_{2}(M, ∂M; ℤ) then x(α + β) ≤ x(α) + x(β).

Proof: Represent each class α, β by PL-embedded oriented surfaces A, B. We can assume that these surfaces intersect transversally: their intersection is a 1-dimensional submanifold Γ. By Theorem 2.25 we can assume that each surface A, B is incompressible. Consider components of A − Γ and B − Γ. If a component of, say, A − Γ is a disk D_{A} bounded by a loop γ ⊂ A then γ bounds a disk D_{B} on B and we can “trade these disks”: take A := A − D_{A} ∪ D_{B}, B := B − D_{B} ∪ D_{A} (see Figure 2).

By pushing new surfaces apart near γ we eliminate the intersection along γ. This procedure preserves homology classes of A and B (since M is aspherical and the sphere D_{A} ∪ D_{B} is contractible in M) and keeps the Euler characteristics of the surfaces. Thus we may assume that each component of A − Γ and B − Γ is different from the disk. The union of surfaces A∪B with the natural orientation of simplices on A, B represents the class

Figure 2: Trading disks and pushing surfaces apart.

α + β. Now we remove Γ from both A and B and glue the components of the complement as follows. Suppose that C_{A} ⊂ A − Γ is adjacent to a loop γ ⊂ Γ. There are exactly two components of B − Γ ∩ Nbd(γ) which are adjacent to γ. We glue C_{A} to those which induces orientation on γ opposite to the orientation induced by C_{A} and do the same with component C_{B} ⊂ B − Γ (exactly the same way as we did with disks D_{A}, D_{B} above, see Figure 2). We repeat this procedure for all components of A∪B −Γ. The resulting singular 2-cycle R(A ∪ B) has non-transversal self-intersections along Γ. Clearly R(A ∪ B) has the same homology class as α + β. Thus we push the self-intersections apart and the result is an embedded oriented surface A ⊕ B which represents the homology class α + β.

Direct calculation shows that χ(A) + χ(B) = χ(A ⊕ B). Also, no component of A ⊕ B is a sphere. This proves Lemma. □

The Lemma and linearity of the function x on rays implies that x is convex. Finally, the norm x is nondegenerate since there are no elements of H_{2}(M, ∂M; ℤ) − {0} which are represented by tori. □

We conclude that the norm x defined over ℤ extends from H_{2}(M, ∂M; ℤ) to H_{2}(M, ∂M; ℝ) (see Theorem 3.2).

Theorem 3.13 Suppose that dim(H_{2}(M, ∂M; ℝ)) ≥ 2. Then for any ζ ∈ H_{2}(M, ∂M; ℤ) such that v = ζ/x(ζ) is a vertex of the polyhedron B^{(x)}(1), ζ can not be represented by a fiber in a fibration of M over 𝕊^{1}.

Proof: Suppose that ζ is a class in H_{2}(M, ∂M; ℤ) represented by a fibration of M over 𝕊^{1}. Recall that such fibration gives a nonzero element τ ∈ H_{2}(M, ∂M)^{∗}. We proved that there exists a positive number ϵ such that the linear function τ satisfies the following properties:

• The linear function τ is integral, i.e. τ : H_{2}(M, ∂M; ℤ) → ℤ;

• for all integral classes ξ ∈ H_{2}(M, ∂M; ℤ) such that

∥ξ/x(ξ) − v∥ ≤ ϵ

we have: x(ξ) = −τ(ξ).

Rational classes are dense in the cone

C_{ϵ}(v) = {α ∈ H_{2}(M, ∂M; ℝ) : ∥α/x(α) − v∥ ≤ ϵ}

Thus we conclude that in this cone the integral linear function −τ coincides with the Thurston’s norm x. Therefore the unit vector v belongs to the interior of a top-dimensional face of the unit sphere ∂B^{(x)}(1), which is locally (near v) is given by the equation τ(β) = −1. This contradicts our assumptions. □

Now we can prove the main theorem of this chapter (Theorem 3.1).

Proof: Indeed, take any vertex v of the polyhedron B^{(x)}(1). According to Corollary 3.6 there exists an integral class ζ ∈ H_{2}(M, ∂M; ℤ) such that v = ζ/x(ζ). On the other hand, by Theorem 3.13 the class ζ is not represented by a fiber in any fibration of M over 𝕊^{1}. The class ζ is nontrivial, thus it can be represented by a superincompressible surface S according to Theorem 2.25. □

4 Geometry of the Hyperbolic Space

4.1 General definitions and notations

Suppose that X is a metric space and A is a subset. We shall use the notation Nbd_{t}(A) to denote the t-neighborhood of A in X. Recall that the Hausdorff distance between two closed subsets A, B of X is

dist_{H}(A, B) = inf{t : A ⊂ Nbd_{t}(B), B ⊂ Nbd_{t}(A)}

This distance can be infinite. The minimal distance between A, B is

d(A, B) = inf{d(a, b) : a ∈ A, b ∈ B}

We shall assume that geodesics and geodesic rays are parameterized by the unit speed.

Suppose that X is a topological space, C(X) is the collection of closed subsets of X. We topologize C(X) as follows. We say that a sequence C_{n} ∈ C(X) is convergent to C ∈ C(X) in the Chabauty topology if

• For every point x ∈ C there exists a sequence x_{n} ∈ C_{n} such that lim_{n→∞} x_{n} = x;

• For every sequence of points x_{n} ∈ C_{n} which is convergent in X we have: lim_{n→∞} x_{n} ∈ C.

In the case when X is a locally-compact metric space one can describe the Chabauty topology as follows. A sequence C_{n} ∈ C(X) is convergent to C ∈ C(X) in the Chabauty topology if for every metric ball B_{r}(x_{0}) ⊂ X we have:

lim_{n→∞} dist_{H}(C_{n} ∩ B_{r}(x_{0}), C ∩ B_{r}(x_{0})) = 0

Suppose that M is a complete Riemannian manifold. Then the injectivity radius InRad_{x}(M) of M at a point x ∈ M is the smallest number r such that M contains a nontrivial geodesic with the both end-points at x and length 2r. Another way to define it is to say that the exponential map exp_{x} : T_{x}M → M is injective on the open ball of radius r and is not injective on its closure. The injectivity radius of M is defined as

InRad(M) = inf_{x∈M} InRad_{x}(M)

We say that a subset C ⊂ M of a metric length space M is convex if any distance-minimizing geodesic g ⊂ M connecting points of C is contained in C.

4.2 CAT (λ)-spaces

The term “CAT (λ)-space” was (I believe) invented by Gromov in his essay [82] and has nothing to do with cats: CAT stands for Cartan-Alexandrov-Toponogov. I think that historically correct term would be RAT (λ): Rauch-Alexandrov-Toponogov. However the term CAT (.) is already widely used and, besides, who likes rats anyway...

I give the definition in the case λ ≤ 0. A metric space is called geodesic if any pair of points can be connected by a geodesic segment. A complete simply-connected geodesic metric space X is called CAT (λ) if it satisfies the conclusion of Rauch-Toponogov comparison theorem. Namely, for any geodesic triangle ABC ⊂ X place a point D on the side [AC]. Take a simply-connected complete Riemannian 2-manifold M(λ) of the constant curvature λ. Draw in M(λ) a comparison triangle ĀB̄C̄ which has the same side-lengths as

Figure 3: Comparison triangles

the original triangle and place a point D̄ ∈ [ĀC̄] within the distance d(A, D) from Ā. Then d(B, D) ≤ d(B̄, D̄). See Figure 3.

Basic examples of CAT (λ)-spaces:

(1) Any simply-connected complete Riemannian manifold of sectional curvature ≤ λ. This is a corollary of Rauch and Toponogov comparison theorems.

(2) Suppose that S is a 2-dimensional cell-complex. Assume that d is a geodesic metric on S whose restriction to each cell is a Riemannian metric of the constant curvature λ so that the edges of S are geodesic. Then we can define total angles around vertices of S by adding up angles in the adjacent corners of adjacent cells. These angles determine a metric on each connected component of the link of each vertex of S. We require each embedded topological circle in the link of each vertex to have length at least 2π. Then the universal cover of S is CAT (λ). The reader can find the proof in [10].

(3) Suppose that X, Y are CAT (0)-spaces, then their direct product X × Y is again CAT (0)-space. I recall that the metric on the direct product is defined by the formula:

d((x, y), (x′, y′))^{2} = d(x, x′)^{2} + d(y, y′)^{2}

(4) Suppose that X is a CAT (−∞)-space. Then X is a metric tree; indeed any geodesic triangle in X must degenerate to a tripod.

The distance function on CAT (0)-spaces is convex in the following sense: take two geodesics γ_{1}(t), γ_{2}(t) in CAT (0)-space which are parameterized by arc-length. Then the function f(t) = d(γ_{1}(t), γ_{2}(t)) is convex.

Exercise 4.1 Show that metric balls in CAT (0)-spaces are strictly convex.

Proposition 4.2 Suppose that X is a CAT (0)-space, F is a finite group of isometries of X. Then F has a fixed point in X.

Proof: Pick a point x ∈ X and consider the orbit F · x = {x_{1}, ..., x_{n}}. Define the function

r(y) = max_{j=1}^{n} d(y, x_{j})

This function is strictly convex, hence if it has a minimum in X, this minimum must be unique. Existence of the minimum follows from completeness of X. Therefore the point of minimum is invariant under the group F. □

The ideal boundary Any CAT (0)-space X has an ideal boundary. First we describe this boundary geometrically. Choose a base-point x_{0} ∈ X. Then points at infinity ξ ∈ ∂_{∞}(X, x_{0}) are rays r_{ξ} : [0, ∞) → X (parameterized as usual by their arc-length) emanating from x_{0}. We will refer to them as rays going to the points ξ at infinity (or asymptotic to the points ξ at infinity) and ∂_{∞}(X, x_{0}) as the ideal boundary of (X, x_{0}).

If X is a Riemannian manifold of nonpositive curvature then topology on ∂_{∞}(X, x_{0}) is defined via the angles between corresponding rays: two points at infinity are close iff the angle between rays is small. In the general case the topology can be defined as follows. Let ρ_{1}, ρ_{2} are rays emanating from x_{0}. For ϵ > 0 we say that the corresponding points at infinity ξ_{1}, ξ_{2} are ϵ-close if d(ρ_{1}(1/ϵ), ρ_{2}(1/ϵ)) ≤ ϵ. We say that a point x ∈ X is ϵ-close to ξ ∈ ∂_{∞}(X, x_{0}) (which is represented by a ray ρ) if d(x, ρ(ℝ_{+})) ≤ ϵ. If X is locally compact (as in the case of the hyperbolic n-spaces), then X̄ = X ∪ ∂_{∞}X is compact.

If we have two distinct base-points x_{0}, x_{1}, then we identify the ideal boundaries ∂_{∞}(X, x_{0}), ∂_{∞}(X, x_{1}) as follows. Suppose ρ_{0}, ρ_{1} are geodesic rays emanating from x_{0}, x_{1}. Then ρ_{0} ∼ ρ_{1} if

d(ρ_{0}(t), ρ_{1}(t)) ≤ C

for some constant C which depends on ρ_{0}, ρ_{1}. This gives a natural identification between the ideal boundaries. It is an easy exercise to verify that this identification is a homeomorphism. With this in mind we will drop the base-point label in ∂_{∞}(X, x_{1}) and will use the notation ∂_{∞}X instead.

Let ρ : [0, ∞) → X be a geodesic ray which is asymptotic to ξ. The Busemann function β_{ξ} corresponding to ρ is the monotone limit :

β_{ξ}(z) := lim_{t→∞}[d(z, ρ(t)) − t]

for z ∈ X. It is easy to see that the function β_{ξ} is a convex function and is well-defined up to an additive constant. For every ray ρ asymptotic to ξ we have:

β_{ξ} ∘ r(t) = −t + const

Thus we may identify points of ∂_{∞}X with the Busemann functions on X (considered up to additive constants). For each point x ∈ X consider the distance function

d_{x} : z ↦ d(z, x)

Now choose a base-point x_{0} ∈ X and normalize d_{x} as:

α_{x}(z) := d_{x}(z) − d(x_{0}, x)

Then α_{x}(x_{0}) = 0 and its sublevel sets are metric balls in X. So we have a homeomorphic embedding

ι : X ↪ C(X, ℝ), x ↦ α_{x}

from X to the space of continuous functions on X (provided with the topology of pointwise convergence). The limit of the functions α_{x} as x → ξ ∈ ∂_{∞}X is the Busemann function β_{ξ} (normalized at x_{0}). Hence the topology on X̄ is the topology induced on the closure of ι(X) in C(X, ℝ).

Horoballs in X are sublevel sets of Busemann functions. Hence each horoball is approximated by a sequence of metric balls in the Chabauty topology.

4.3 The basic properties of ℍ^{n}

The n-dimensional hyperbolic space ℍ^{n} is the complete simply-connected Riemannian manifold of the constant sectional curvature −1. I recall the basic properties of this space.

• The Gauss-Bonnet formula: If [abc] is a geodesic triangle in ℍ^{n} then the sum of angles in [abc] equals π − Area([abc]).

• Thin triangles property: If Δ is any geodesic triangle in ℍ^{n}, then each side of Δ is contained in the 2-neighborhood of the union of two other sides.

• The distance function is convex.

• The distance function to convex sets is convex. Suppose that C ⊂ ℍ^{n} is a nonempty convex subset, r > 0, then Nbd_{r}(C) is strictly convex.

• The space ℍ^{n} has no conjugate points: any two complete distinct geodesics intersect in at most one point.

Lemma 4.3 Consider a “twisted quadrilateral” in ℍ^{n}: a configuration of four points A, B, C, D in ℍ^{n} such that

∠ABC ≥ π/2, ∠DAB = π/2, ∠ADC ≥ π/2

and d(D, A) ≥ h. Then sinh d(A, B) ≤ 1/ sinh h.

Proof: First, consider a quadrilatteral [A′B′C′D′] ⊂ ℍ^{2} so that

∠A′B′C′ = ∠A′D′C′ = ∠D′A′B′ = π/2

Then

sinh(d(A′, B′)) sinh(d(A′, D′)) = cos(∠D′C′B′) ≤ 1

(see [13, Theorem 7.17.1]). Clearly, the same conclusion holds if we only assume that

∠A′B′C′, ∠A′D′C′ ≥ ∠D′A′B′ = π/2

(See Figure 4.) Now draw the diagonal [DB] in the quadrilatteral [ABCD] and rotate the triangle [DAB] around the line [DB], keeping the triangle [DCB] fixed until we get a quadrilatteral [A′B′C′D′] embedded in a hyperbolic plane ℍ^{2} ⊂ ℍ^{n} which has the same properties as the original one since ∠D′A′B′ = ∠DAB and ∠A′B′C′ ≥ ∠ABC, ∠A′D′C′ ≥ ∠ADC. □

Figure 4: Twisted rectangle

The following lemma shows that the nearest-point projections in ℍ^{n} decrease length exponentially.

Lemma 4.4 Suppose that γ is a geodesic in ℍ^{n} and Q ⊂ ℍ^{n} is a closed convex subset such that d(γ, C) ≥ h > 0.2. Then the diameter of the nearest-point projection of γ to C is at most 4 exp(−h).

Proof: Suppose that A ∈ Q, D ∈ γ realize the minimal distance between Q and γ. Let C ∈ γ − {D} be any point and B ∈ Q be its projection. Now we apply Lemma 4.3 to the configuration A, B, C, D:

d(A, B) ≤ sinh(d(A, B)) ≤ 1/ sinh(h)

However, if h ≤ 0.2 then 1/ sinh(h) ≤ 4e^{−h}. □

Corollary 4.5 Suppose that ρ_{1}(t), ρ_{2}(t) are geodesic rays in ℍ^{n} which determine the same point ξ at infinity. Then there is a constant c so that

|ρ_{1}(t) − ρ_{2}(t − c)| ∼ exp(−t)

as t → ∞.

The constant c is chosen so that the points ρ_{1}(t), ρ_{2}(t − c) belong to the same horosphere with center at ξ.

4.4 Models of the hyperbolic space

There are two basic models of the hyperbolic space that we shall use. The first one is the “upper half-space” model. Let ℍ^{n} = ℝ_{+}^{n} = {x : x_{n} > 0}; then the hyperbolic metric is ds = |dx|/x_{n}. Geodesics in this model are circular arcs and Euclidean rays in ℝ_{+}^{n} which are orthogonal to the boundary ℝ^{n−1} ∪ {∞}. For instance, let x, y ∈ ℍ^{n} be points with coordinates (0, …, 0, t), (0, …, 0, s); then d(x, y) = | log(s/t)|.

The other model is the unit ball model. Namely, let ℍ^{n} = 𝔹^{n} = {x ∈ ℝ^{n} : |x| < 1} with the metric

ds = \frac{|dx|}{1 − |x|^{2}}

Geodesics are Euclidean segments and circular arcs which are orthogonal to the boundary 𝕊^{n−1}.

In the both models closed metric balls in ℍ^{n} are closed Euclidean balls which are contained in 𝔹^{n} and ℝ_{+}^{n} respectively. Angles in ℍ^{n} are the usual angles in ℝ^{n} since the hyperbolic metric is conformally-Euclidean.

The space ℍ^{n} has the natural compactification ℍ̄^{n} = ℍ^{n} ∪ ∂_{∞}ℍ^{n} defined in Section 4.2. Here we describe this compactification using models of ℍ^{n}. Identify ℍ^{n} with the unit ball 𝔹^{n} with the center x_{0} := 0. Then geodesic rays emanating from the origin are the Euclidean segments, thus the compactification of ℍ^{n} is naturally homeomorphic to the closure 𝔹̄^{n} of the unit ball in ℝ^{n}.

If we identify ℍ^{n} with the upper half-space then the compactification ∂_{∞}ℍ^{n} ∪ ℍ^{n} is the closure \overline{ℝ_{+}^{n}} = \{x ∈ ℝ^{n} : x_{n} ≥ 0\} ∪ \{∞\}.

In any case the set of points at infinity is homeomorphic to the sphere 𝕊_{∞}^{n−1} which we call the sphere at infinity of the hyperbolic space. We give 𝕊_{∞}^{n−1} the “spherical metric” (of the constant curvature +1). For n = 3 (the case we are mainly interested in) we shall identify the sphere at infinity with the extended complex plane \overline{ℂ}.

Horoballs in ℍ^{n} are represented by Euclidean balls in 𝔹^{n} which are tangent to ∂_{∞}ℍ^{n}. Boundaries of horoballs are called horospheres. Intrinsically horoballs are limits of metric balls in ℍ^{n} whose radius tends to infinity (in the Chabauty topology). Another point of view is that horoballs are sublevel sets of Busemann functions in X, see Section 4.2.

4.5 Isometries of the hyperbolic space

The easiest way to understand the group of isometries Isom(ℍ^{n}) of ℍ^{n} is to consider different models of the hyperbolic space. We start with the unit ball model 𝔹^{n}. Then the hyperbolic metric is rotationally-symmetric and thus K := O(n) ⊂ Isom(ℍ^{n}) is the stabilizer of the origin 0 in Isom(ℍ^{n}). The group O(n) is a maximal compact subgroup in Isom(ℍ^{n}). Now we move to the upper half-space model ℝ_{+}^{n}. Let B be the group which is generated by Euclidean translations in ℝ^{n} that keep ℝ_{+}^{n} invariant and Euclidean dilations \overrightarrow{x} ↦ k\overrightarrow{x}, k > 0.

Direct calculations show that B belongs to the group Isom(ℍ^{n}). In particular, the group Isom(ℍ^{n}) acts transitively on ℍ^{n}. Moreover, for each two points ξ, η in the unit tangent bundle U(ℍ^{n}), there is an isometry g : ξ ↦ η. However any isometry of a connected Riemannian manifold M, which fixes an element of the unit tangent bundle ξ ∈ U(M), must be the identity itself. Thus: Isom(ℍ^{n}) = K · B, i.e. any element g ∈ Isom(ℍ^{n}) splits as a composition g = k ∘ b, where k ∈ K, b ∈ B.

The group of isometries Isom(ℍ^{n}) of ℍ^{n} is isomorphic to the Lorentz group SO(n, 1); if n = 3 then Isom_{+}(ℍ^{3}) = PSL(2, ℂ) is the group of orientation-preserving isometries. In general the group Isom(ℍ^{n}) acts on 𝕊_{∞}^{n−1} as the group of Moebius transformations. For each isometry g ∈ Isom(ℍ^{n}) we define the translational length ℓ(g):

ℓ(g) = inf_{x∈ℍ^{n}} d(x, g(x))

There are three types of conjugacy classes of nontrivial isometries of ℍ^{n}:

• loxodromic: ℓ(g) > 0,

• parabolic: ℓ(g) = 0 but g does not have a fixed point in ℍ^{n},

• elliptic: g fixes a point in ℍ^{n}.

Any loxodromic element h has a unique invariant geodesic A_{h} in ℍ^{n}. This invariant geodesic is called the axis of h. The restriction of h to A_{h} is a nontrivial translation. The translational length ℓ(h) of a loxodromic element h is equal to the length of A_{h}/⟨h⟩. Loxodromic elements have exactly two fixed points in \overline{ℍ^{n}}. One of these points is called repulsive and another is called attractive. The difference is that under iterations of h any point z ∈ A_{h} moves towards the attractive fixed point and away from the repulsive point (the same is actually true for all z ∈ ℍ^{n}). A loxodromic element g is called hyperbolic if it is conjugate to the dilation

\overrightarrow{x} ↦ k\overrightarrow{x}

where k > 0, in the upper half-space model of ℍ^{n}. Geometrically this means that g preserves any totally-geodesic half-plane with the boundary A_{g}.

Parabolic elements h have exactly one fixed point in \overline{ℍ^{n}} and thus they have no invariant geodesics. Any parabolic element is conjugate to a Euclidean isometry in the group B. Any elliptic element g ∈ Isom(ℍ^{n}) has a nonempty totally geodesic submanifold which is pointwise fixed by g. Any elliptic element is conjugate to an element of the group K. For g ∈ Isom(ℍ^{n}) let F ix_{X}(g) denote the fixed–point set for the action of g on a subset X ⊂ ℍ^{n} ∪ ∂ℍ^{n}.

In the case of n = 3 each orientation-preserving isometry corresponds to an element of P SL(2, ℂ), hence we can define its trace up to the ± sign. Elements of P SL(2, ℂ) are classified as follows:

• Elliptic: T race(g) ∈ (−2, 2); • Parabolic: T race(g) = ±2;

• Loxodromic: T race(g) ∉ [−2, 2].

Exercise 4.6 Let g ∈ SL be a loxodromic element with the trace t. Then

t = λ + λ^{−1}, ℓ(g) = | log(|λ|)|

Hint: verify this formula in the case of loxodromic elements of the form g(z) = λ^{2} · z.

Let g be a nonelliptic element of Isom(ℍ^{3}), T = ℍ^{3}/⟨g⟩. Given ϵ > 0 we use the notation T_{(0,ϵ]} for the subset of T where the injectivity radius is at most ϵ/2.

Lemma 4.7 T_{(0,ϵ]} is convex.

Remark 4.8 Convexity fails for hyperbolic spaces of dimension ≥ 4.

Proof: Let X be the lift of T_{(0,ϵ]} to the universal cover ℍ^{3}. Clearly it is enough to establish convexity of X. Note that X must be invariant under the centralizer of the group ⟨g⟩ in Isom(ℍ^{3}). If g is loxodromic we can assume that g : z ↦ kz, k ∈ ℂ, |k| ≠ 0, 1, up to conjugation. The centralizer consists of the similarities z ↦ λz, λ ∈ ℂ^{∗}. Thus the boundary of X is a Euclidean cone in ℍ^{3} with the axis A_{g} and X is a hypercycle. Convexity of X follows from the description of geodesics in ℍ^{3}. If h is parabolic then g : z ↦ z + 1 and the group z ↦ z + a, a ∈ ℂ is the centralizer of ⟨g⟩. Hence the boundary of X is a horosphere with the center at the point ∞. Convexity of X follows. □

If g is loxodromic let γ be the simple closed geodesic A_{g}/⟨g⟩ in T. Then T_{(0,ϵ]} ∪ γ is nonempty. Let q ⊂ T = ℍ^{3}/⟨g⟩ be a piecewise–geodesic loop homotopic to [g] which consists of r geodesic segments.

Lemma 4.9 Under conditions above we have:

d(q, T_{(0,ϵ]} ∪ γ) ≤ 0.2 − log(ϵ/2) + 2r

Proof: Let x ∈ q be one of the vertices, connect this point to itself by a geodesic segment α in T which is homotopic to q (rel {x}). Then the resulting loop is contained in the 2rneighborhood of q (by the “thin triangles” property of the hyperbolic space). We lift α in ℍ^{3} to a geodesic segment β which connects points y, g(y), where y covers x. Suppose that T_{(0,ϵ]} ≠ ∅. Let h be the minimal distance between the lift C of T_{(0,ϵ]} into ℍ^{3} and β. The set C is closed and convex, therefore the length of the nearest–point projection π_{C}(β) is at most max(4 exp(−h), 0.2) and the end–points of π_{C}(β) are identified by the element g. Hence

ϵ ≥ 4 exp(−h) or h ≤ 0.2

which means that h ≤ 0.2−log(ϵ/4). We conclude that the minimal distance between q and T_{(0,ϵ]} is at most 0.2 − log(ϵ/4) + 2r. The same argument works in the case when T_{(0,ϵ]} = ∅, which implies ℓ(g) > ϵ. In this case we consider the projection of β to the axis of g. □

Let g be a loxodromic isometry of ℍ^{n}. The quotient ℍ^{n}/⟨g⟩ contains the closed geodesic γ = A_{g}/⟨g⟩. Suppose that q ⊂ ℍ^{n}/⟨g⟩ is a piecewise-geodesic loop which is freely homotopic to γ and has a single vertex z. Let d(γ, q) = h.

Lemma 4.10 Under the conditions above the closed geodesic γ is contained in the (h + 4)- neighborhood of the loop q.

Figure 5:

Proof: Lift the curve q to the universal cover ℍ^{n} and choose a point A in the lift which is nearest to A_{g}. Let B = g(A) and A′, B′ ∈ A_{g} be the nearest points to A, B. Then the projection of the segment [A′B′] to ℍ^{n}/⟨g⟩ is the geodesic loop γ. (See Figure 5). The “thin triangle property” (see Section 4.3) implies that [A′B′] is within (4 + h)-neighborhood of the lift of q. □

4.6 Convergence property for Moebius transformations

Proposition 4.11 The group G = Isom(ℍ^{n}) has the Cartan decomposition: G = KAK.

Proof: G = KAK means the following. Pick a base-point O ∈ ℍ^{n}, let K be the stabilizer of O in G, i.e. K is a maximal compact subgroup of G. Choose a geodesic L ⊂ ℍ^{n} passing through O, let A be the group which consists of the identity and of hyperbolic translations with the axis L. Then G = KAK means that for each g ∈ G there are elements k, k′ ∈ K and a ∈ A so that g = kak′.

Recall that the group G acts transitively on ℍ^{n} and we can identify the quotient G/K with the hyperbolic space ℍ^{n}, so that 1 projects to the point O. Let g · K = x ∈ ℍ^{n} be any point. Then the orbit K(x) is the metric sphere with center at O passing through x. In particular we can find an element k ∈ K so that k(x) ∈ L. Then take a translation a ∈ A so that ak(x) = O. Thus a · k · g · K = K and g ∈ KAK. □

For a pair of points z, x ∈ 𝕊^{n−1} we define a quasiconstant map z_{x} : 𝕊^{n−1} → 𝕊^{n−1} so that: z_{x}(w) = z for each w ≠ x, and z_{x} is undefined at x. The group G = Isom(ℍ^{n}) acts on 𝕊^{n−1} as the group of Moebius transformations. A sequence of elements g_{n} ∈ G ↷ 𝕊^{n−1} is convergent to a quasiconstant map z_{x} iff g_{n} converges to z_{x} uniformly on compacts in 𝕊^{n−1} − {x}. Note that g_{n} → z_{x} implies g_{n}^{−1} → x_{z}. The space of quasiconstants Q(𝕊^{n−1}) is naturally homeomorphic to 𝕊^{n−1} × 𝕊^{n−1}. Thus we have a topology on Ĝ = G ∪ Q(𝕊^{n−1}).

Proposition 4.12 (Convergence property for Moebius transformations.) Ĝ = G ∪ Q(𝕊^{n−1}) is compact.

Proof: Recall that for the group G we have the Cartan decomposition: G = KAK, defined above. Let ξ, η ∈ 𝕊_{∞}^{n−1} be the end-points of the geodesic L. Hence for any sequence g_{i} ∈ G we have: g_{i} = k_{i}a_{i}c_{i}, where k_{i}, c_{i} ∈ K, a_{i} ∈ A. The sequences k_{i}, c_{i} are subconvergent to k ∈ K and c ∈ K; a_{i} is subconvergent to a, where a is either an element of A or is one of the quasiconstants ξ_{η}, η_{ξ}. Suppose that a = η_{ξ}. Then g_{i} is subconvergent on 𝕊^{n−1} − c^{−1}(ξ) to k(η). Thus g_{i} → x_{y}, where x = k(η), y = c^{−1}(ξ). □

4.7 Convex polyhedra in the hyperbolic space

Suppose that Φ is a locally finite convex polyhedron in ℍ^{n} where all dihedral angles between faces are ≤ π/2. In this case we will say that Φ has acute angles. For each top-dimensional face F_{j} of Φ let F̂_{j} denote the hyperplane in ℍ^{n} which contains F_{j}. The half-space in ℍ^{n} bounded by F̂_{j} and disjoint from Φ will be denoted by F̂_{j}^{−}.

Theorem 4.13 (E. Andreev, [8]) If Φ has only acute angles then:

faces F_{i}, F_{j} of Φ intersect if and only if F̂_{j} intersects F̂_{i} in ℍ^{n};

For n = 3 we have

Theorem 4.14 Faces F_{i}, F_{j}, F_{k} of Φ intersect if and only if F̂_{j}, F̂_{i}, F̂_{k} intersect in ℍ^{3}.

Proof: Suppose that F is a face of Φ so that F̂ separates O := F̂_{1} ∩ F̂_{2} ∩ F̂_{3} from the polyhedron Φ. (Figure 6.) Our goal is to show that this is impossible. If such configuration of hyperplanes exists then there is a simplex T ⊂ ℍ^{3} whose vertices are O, F̂_{1} ∩ F̂_{2} ∩ F̂, F̂_{1} ∩ F̂_{3} ∩ F̂ and F̂_{2} ∩ F̂_{3} ∩ F̂. The dihedral angles of this simplex satisfy the properties:

• All dihedral angles at the vertex O are ≤ π/2;

• All dihedral angles at the face of T opposite to O are ≥ π/2.

Let p, q, r denote the dihedral angles of T at the vertex O; x, y, z denote the dihedral angles at the opposite face.

The cosine formula of the spherical trigonometry implies that the planar angle ∠OBC is obtuse unless cos z − cos y cos r > 0 and ∠OBA > π/2. Since hyperbolic triangle can not have two obtuse angles we conclude that (up to change of notations):

cos z − cos y cos r > 0
cos y − cos x cos q > 0
cos x − cos z cos p > 0

Since cos p, cos q, cos r ≥ 0 it implies that cos r cos p cos q ≥ 1 which is impossible. □

Corollary 4.15 Suppose that F_{1}, F_{2}, F_{3}, F_{4} are distinct faces of Φ. Then the intersection

F̂_{1}^{−} ∩ F̂_{2}^{−} ∩ F̂_{3}^{−} ∩ F̂_{4}^{−}

is always empty.

4.8 Cayley graphs of finitely generated groups

Let Γ be a finitely generated group with the generating set S = {s_{1}, ..., s_{n}}, we shall assume that the identity does not belong to S and S = S^{−1}, i.e. for each s_{i} ∈ S, the element s_{i}^{−1} also belongs to S. Define the Cayley graph C = C(Γ, S) as follows: the vertices of C are the elements of Γ. Two vertices g, h ∈ Γ are connected by an edge if an only if there is a generator s_{i} ∈ S such that h = gs_{i}. Then C is a locally finite graph. Define the word metric d on C by assuming that each edge has the unit length, this defines length of finite PL-paths in C, finally the distance between points p, q ∈ C is the infimum (same as minimum) of the lengths of PL-paths in C conecting p to q. For g ∈ G the word length ℓ(g) is just the distance d(1, g) in C. It is clear that the left action of the group Γ on the metric space (C, d) is isometric.

Below are two simple examples of Cayley graphs.

Example 4.16 Let Γ be free abelian group on two generators s_{1}, s_{2}. Then S = {s_{i}^{±1}, i = 1, 2}. The Cayley graph C = C(Γ, S) is the square grid in the Euclidean plane: the vertices are points with integer coordinates, two vertices are connected by an edge if and only if only two of their coordinates are distinct and they differ by ±1.

Example 4.17 Let Γ be the free group on two generators s_{1}, s_{2}. Then S = {s_{i}^{±1}, i = 1, 2}. The Cayley graph C = C(Γ, S) is the 4-valent tree (there are four edges incident to each vertex).

4.9 Quasi-isometries and quasi-conformal homeomorphisms

Definition 4.18 Let X, Y be complete metric spaces. A map f : X → Y is called a (k, c)- quasi-isometric embedding if

k^{−1}d(x, x′) − c ≤ d(f(x), f(x′)) ≤ kd(x, x′) + c  (5)

for any x, x′ ∈ X.

A (k, c)-quasi-isometric embedding is called a (k, c)-quasi-isometry if it admits an approximate inverse map f̄: Y → X which is a (k, c)-quasi-isometric embedding so that:

d(f̄f(x), x) ≤ c, d(ff̄(y), y) ≤ c  (6)

for each x ∈ X, y ∈ Y.

In most cases the quasi-isometry constants k, c do not matter, so we shall use the terms quasi-isometries and quasi-isometric embeddings without specifying constants. Note that a quasi-isometric embedding does not have to be an embedding in the usual sense, what is true however is that distant points must have distinct images.

Exercise 4.19 If f : X → Y is a quasi-isometry and g is within finite distance from f (i.e. sup d(f(x), g(x)) < ∞) then g is also a quasi-isometry.

Exercise 4.20 A subset S of a metric space X is said to be r-dense in X if the Hausdorff distance between S and X is at most r. Show that if f : X → Y is a quasi-isometric embedding such that f(X) is r-dense in X for some r < ∞ then f is a quasi-isometry. Hint: construct quasi-inverse f̄ to the map f by mapping point y ∈ Y to x ∈ X such that

d(f(x), y) ≤ d(f(X), y) + 1

Definition 4.21 A geodesic metric space is a metric space where any two points can be connected by a geodesic path. Proper metric space is a space where closed metric balls are compact.

Lemma 4.22 Let X be a proper geodesic metric. Let G be a group acting properly discontinuously cocompactly on X. Pick a point x_{0} ∈ X. Then the group G is finitely generated and for some choice of finite generating set S of the group G ,the map f : G → X, given by f(g) = g(x_{0}), is a quasi-isometry. Here G is given the word metric induced from C(G, S).

Proof: Our proof follows [74, Section 10]. Let B = B_{R}(x_{0}) be the closed ball of radius R in X with the center at x_{0} such that B_{R−1}(x_{0}) projects onto X/G. Since the action og G is properly discontinuous, there are only finitely many elements s_{i} ∈ G − {1} such that B ∩ s_{i}B ≠ ∅. Let S be the subset of G which consists of the above elements s_{i} (it is clear that s_{i}^{−1} also belongs to S iff s_{i} does). Let

r := inf{d(B, g(B)), g ∈ G − (S ∪ {1})}

Clearly r > 0. We claim that S is a generating set of G and that for each g ∈ G

ℓ(g) ≤ d(x_{0}, g(x_{0}))/r + 1  (7)

where ℓ is the word length on G (with respect to the generating set S). Let g ∈ G, connect x_{0} to g(x_{0}) by the shortest geodesic γ. Let m be the smallest integer so that d(x_{0}, g(x_{0})) ≤ mr+R. Choose points x_{1}, ..., x_{m+1} = g(x_{0}) ∈ γ, so that x_{1} ∈ B, d(x_{j}, x_{j+1}) < r, 1 ≤ j ≤ m. Then each x_{j} belongs to g_{j}(B) for some g_{j} ∈ G. Let 1 ≤ j ≤ m, then g_{j}^{−1}(x_{j}) ∈ B and d(g_{j}^{−1}(g_{j+1}(B)), B) ≤ d(g_{j}^{−1}(x_{j}), g_{j}^{−1}(x_{j+1})) < r. Thus the balls B, g_{j}^{−1}(g_{j+1}(B)) intersect which means that g_{j+1} = g_{j}s_{i(j)} for some s_{i(j)} ∈ S ∪ {1}. Therefore

g = s_{i(1)}s_{i(2)}....s_{i(m)}

We conclude that S is indeed a generating set for the group G. Moreover,

ℓ(g) ≤ m ≤ (d(x_{0}, g(x_{0})) − R)/r + 1 ≤ d(x_{0}, g(x_{0}))/r + 1

The word metric on the Cayley graph C = C(G, S) of the group G is left-invariant, thus for each h ∈ G we have:

d(h, hg) = d(1, g) ≤ d(x_{0}, g(x_{0}))/r + 1 = d(h(x_{0}), hg(x_{0}))/r + 1

Hence for any g_{1}, g_{2} ∈ G

d(g_{1}, g_{2}) ≤ d(f(g_{1}), f(g_{2}))/r + 1

On the other hand, the triangle inequality implies that

d(x_{0}, g(x_{0})) ≤ tℓ(g)

where d(x_{0}, s(x_{0})) ≤ t ≤ 2R for all s ∈ S. Thus

d(f(g_{1}), f(g_{2}))/t ≤ d(g_{1}, g_{2})

We conclude that the map f : G → X is a quasi-isometric embedding. Since f(G) is R-dense in X, it follows that f is a quasi-isometry. □

Corollary 4.23 Let S_{1}, S_{2} be finite generating sets for a finitely generated group G and d_{1}, d_{2} be word metrics on G corresponding to S_{1}, S_{2}. Then the identity map (G, d_{1}) → (G, d_{2}) is a quasi-isometry.

Proof: The group G acts isometrically cocompactly on the proper metric space (C(G, S_{2}), d_{2}). Thus the map id : G → C(G, S_{2}) is a quasi-isometry. □

Lemma 4.24 Let X be a locally compact path-connected topological space, G is a group acting properly discontinuously cocompactly on X. Let d_{1}, d_{2} be two proper geodesic metrics on X (consistent with the topology of X) both invariant under the action of G. Then the group G is finitely generated and the identity map id : (X, d_{1}) → (X, d_{2}) is a quasi-isometry.

Proof: The group G is finitely generated by Lemma 4.22, choose a word metric d on G corresponding to any finite generating set (according to the previous corollary it does not matter which one). Pick a point x_{0} ∈ X, then the maps

f_{i} : (G, d) → (X, d_{i}), g ↦ g(x_{0})

are quasi-isometries, let f̄_{i} denote their quasi-inverses. Then the map id : (X, d_{1}) → (X, d_{2}) is within finite distance from the quasi-isometry f_{2} ∘ f̄_{1}. □

A (k, c)-quasi-geodesic segment in a metric space X is a (k, c)-quasi-isometric embedding f : [a, b] → X; similarly, a (k, c)-quasi-geodesic is a (k, c)-quasi-isometric embedding f : ℝ → X. By abusing notation we will refer to the image of a (k, c)-quasi-geodesic as a quasi-geodesic.

Corollary 4.25 Let d_{1}, d_{2} be as in Lemma 4.24. Then any (complete) geodesic γ with respect to the metric d_{1} is also a quasi-geodesic with respect to the metric d_{2}.

Suppose that (X_{n}, x_{n}) is a sequence of complete pointed metric spaces (x_{n} are basepoints on X_{n}). We say that (X_{n}, x_{n}) is convergent to the metric space (X, x) in the quasiisometric topology (another term is the Gromov-Hausdorff or geometric topology) if the following is satisfied:

For each ϵ > 0, r > 0 there is a number n_{0} < ∞ such that for every n > n_{0} there exists a (1, ϵ)-quasi-isometry

f_{n} : B_{r}(x_{n}) → B_{r}(x)

Definition 4.26 Suppose that X, Y are metric spaces, f : X → Y is a homeomorphism. The mapping f is called quasiconformal if the function

H_{f}(x) = lim sup_{r→0} \frac{sup\{d(f(z), f(x)) : d(x, z) = r\}}{inf\{d(f(z), f(x)) : d(x, z) = r\}}

is bounded from above in X. A quasiconformal mapping is called K-quasi-conformal if the function H_{f} is bounded from above by K a.e. on X.

More on quasiconformal homeomorphisms see in Sections 4.10, 5.1.

The notion of quasiconformality does not work well in the case when the domain and range are 1-dimensional.

Definition 4.27 A homeomorphism f : ℝ → ℝ is called quasisymmetric if there exists k ≥ 1 so that for all x, a ∈ ℝ

k^{−1} ≤ \frac{|f(x + a) − f(a)|}{|f(x) − f(x − a)|} ≤ k

A homeomorphism f : 𝕊^{1} → 𝕊^{1} is called quasimoebius if there is a pair of Moebius transformations A, B : 𝕊^{1} → \overline{ℝ} such that AfB^{−1} is quasisymmetric.

If D ⊂ ∂_{∞}ℍ^{n} is a subset, we provide D with the restriction of the spherical metric from 𝕊_{∞}^{n−1}. Recall that for a subset X ⊂ ℍ^{n}, ∂_{∞}X denotes the set of accumulation points of X in ∂_{∞}ℍ^{n}.

Theorem 4.28 (V. Efremovich, E. Tihomirova [53], see also [201].) Suppose that X, Y ⊂ ℍ^{n} are two closed convex subsets, f : X → Y is a (k, c)-quasi-isometry. Then f has a continuous extension h : ∂_{∞}X → ∂_{∞}Y which is a quasi-conformal homeomorphism between ∂_{∞}X and ∂_{∞}Y (if n ≥ 3) and quasimoebius (if n = 2).

Proof: Suppose that [ab] ⊂ X is a geodesic segment.

Lemma 4.29 There is a function θ(k, c) such that

d_{H}(f([ab]), [f(a), f(b)]) ≤ θ(k, c)

We will prove this assertion in the Lemma 10.7 as an application of ultralimits. Alternatively the reader can derive this lemma from the exponential contraction property for the nearestpoint projections in ℍ^{n} (Lemma 4.4), compare [192, ??] and [16, Lemma C.1.6].

Now let ℓ be an infinite geodesic ray in X with origin at x_{0}. By Lemma 4.29 for every x_{i} we have:

d(j(x_{i}), [j(x_{0}), j(x_{n})]) ≤ θ(k, c)  (8)

for every x_{i}, x_{n} ∈ ℓ. Therefore, there exists a limit ray ϕ(ℓ) for the segments [j(x_{0}), j(x_{n})] (as d(x_{0}, x_{n}) → ∞). This ray has the property:

d(j(x_{i}), ϕ(ℓ)) ≤ θ(k, c)  (9)

for every x_{i} ∈ ℓ. This means that the map f : ℓ → Y has an extension to the point at infinity ∂_{∞}ℓ → ∂_{∞}ϕ(ℓ). Moreover, for every complete geodesic g ⊂ X there is a geodesic ϕ(g) ⊂ Y such that d(f(g), ϕ(g)) ≤ θ(k, c). In particular, each quasi-geodesic in ℍ^{n} has exactly two accumulation points in ∂_{∞}ℍ^{n}.

Clearly the extension of f to infinity: h : ∂_{∞}X → ∂_{∞}Y is a bijection.

I will verify quasiconformality for n ≥ 3 and leave the case n = 2 to the reader, simultaneously our computation will show that h is a homeomorphism. According to the definition, it is enough to verify quasiconformality at each particular point x with uniform estimates on the function H(x). Thus, after composing h with Moebius transformations we can take x = 0 = h(x); h(∞) = ∞, where we consider the upper half-space model of ℍ^{n}.

Take a Euclidean sphere S_{r}(0) in ℝ^{n−1} with center at the origin. This sphere is the ideal boundary of a hyperplane P_{r} which is orthogonal to the vertical geodesic L ⊂ ℍ^{n}, connecting 0 and ∞. Let x_{r} = L ∩ P_{r}. Let π_{L} : ℍ^{n} → L be the nearest point projection. The hyperplane P_{r} can be characterized by the following equivalent properties:

P_{r} = {w ∈ ℍ^{n} : π_{L}(w) = x_{r}}
P_{r} = {w ∈ ℍ^{n} : d(w, x_{r}) = d(w, L)}

Since quasi-isometric images of geodesics in ℍ^{n} are uniformly close to geodesics, we conclude that

diam[π_{L}(f(P_{r}))] ≤ Const

where Const depends only on the quasi-isometry constants of f. The projection π_{L} extends naturally to ∂_{∞}ℍ^{n}. We conclude:

diam[π_{L}(h(S_{r}(0)))] ≤ Const

Thus h(S_{r}(0)) is contained in a spherical shell

{z ∈ ℝ^{n−1} : ρ_{1} ≤ |z| ≤ ρ_{2}}

where log[ρ_{1}/ρ_{2}] ≤ Const. This implies that the function H(z) is bounded from above by K := exp(Const). We conclude that the mapping h is K-quasiconformal. □

4.10 Length distortion under quasiconformal maps of the hyperbolic space

In the case of domains in ℝ^{n} (n ≥ 2) there is another (analytical) description of quasiconformal mappings. Suppose that D, D′ are domains in ℝ^{n}. A homeomorphism f : D → D′ is called quasiconformal if it has distributional partial derivatives in L_{loc}^{2}(D) and the ratio

R_{x} := ∥f′(x)∥/|J_{f}(x)|

is uniformly bounded from above a.e. in D. Here ∥f′(x)∥ is the operator norm of the derivative f′(x) of f at x. The essential supremum of R_{x} in D is denoted by K_{O}(f) and is called the outer dilatation of f. Similarly one defines the inner dilatation K_{I}(f) = ess sup{L_{x}|x ∈ D}, where

L_{x} := |J_{f}(x)|/l(f′(x))^{n}, l(f′(x)) = min_{|h|=1} |f′(x)h|

Let us compare L_{x} and R_{x}. Clearly it is enough to consider positive-definite diagonal matrices f′(x). Let Λ be the maximal eigenvalue of f′(x) and λ be the minimal eigenvalue. Then l(f′(x)) = λ, ∥f′(x)∥ = Λ, H_{f}(x) = Λ/λ. We will restrict ourselves to the case n = 3, then

R_{x}^{3/2} ≤ L_{x} ≤ R_{x}^{6}, R_{x} ≤ H_{f}(x) ≤ R_{x}^{3}

Two definitions of quasiconformality coincide (see for instance [205]) and we have:

K_{O}(f)^{3/2} ≤ K_{I}(f) ≤ K_{O}(f)^{6}, K_{O}(f) ≤ K_{I}(f) ≤ K_{O}(f)^{3}

Note that quasiconformality of mappings and the coefficients of quasiconformality K(f), K_{O}(f), K_{I}(f) do not change if instead of the Euclidean metric we consider a conformally-Euclidean metric in D.

Theorem 4.30 (See [205, Theorem 11.19].) Suppose that f : ℍ^{n} → ℍ^{n} is a quasiconformal mapping, let k^{−1} = K_{O}(f). Then

tanh \frac{1}{4}d(f(x), f(y)) ≤ 2(tanh \frac{1}{4}d(x, y))^{k}

for every x, y ∈ ℍ^{n}.

This means that there are continuous functions θ(s, t), ζ(s, t) such that:

ζ(K_{O}(f), d(x, y)) ≤ d(f(x), f(y)) ≤ θ(K_{O}(f), d(x, y))
θ(s, t) ∼ 8(t/4)^{1/s} as t → 0

Thus we have a (nonlinear) bound on how quasiconformal mappings of ℍ^{n} distort the hyperbolic distance.

4.11 Harmonic functions

Tubular neighborhoods of geodesic rays define another topology on ℍ^{n} ∪ ∂_{∞}ℍ^{n} which is called the conical topology. Namely, a sequence of points x_{k} ∈ ℍ^{n} converges to a point ξ ∈ ∂_{∞}ℍ^{n} in the conical topology, if there is a geodesic ray r_{ξ} going to the point ξ and a number ϵ such that the sequence x_{k} is contained in the ϵ-neighborhood of r_{ξ} and x_{k} → ξ in the topology defined in Section 4.3. If x_{k} → ξ in the conical topology then we shall write this as lim_{k→∞}^{c} x_{k} = ξ

Suppose that λ : ∂_{∞}ℍ^{n} → ℝ is a measurable function which belongs to L^{1}. The function α admits a harmonic extension ext(λ) = f : ℍ^{n} → ℝ which is defined as follows. Pick a point x ∈ ℍ^{n} and consider it as a base-point of ℍ^{n}. This gives an identification of ∂_{∞}ℍ^{n} and S_{x}, which is the unit sphere in the tangent space to ℍ^{n} at x. Then define

f(x) := ext(λ) = ∫_{S_{x}} λ

The function f satisfies the following properties:

• f is harmonic with respect to the Laplace operator of the hyperbolic metric, in particular this is a real-analytic function.

• There is a measurable subset E ⊂ ∂_{∞}ℍ^{n} of full measure such that f ∪ λ : ℍ^{n} ∪ E → ℝ is a continuous in the conical topology.

• ext(λ) ∘ γ = ext(λ ∘ γ) for all γ ∈ Isom(ℍ^{n}).

The function f is called the harmonic extension of λ. For details see [2] , [170].

5 Teichmüller Theory of Riemann surfaces

For detailed discussion of the Teichmüller theory see [1], [73], [123], [157], [199], etc., here I give only brief introduction without proofs.

5.1 Properties of quasiconformal maps

We start our discussion with general properties of quasiconformal homeomorphisms (see Section 4.9 for definition) between domains in 𝕊^{n} (n ≥ 2) and then will restrict ourselves to the case n = 2. Suppose D is a domain in 𝕊^{n}, f : D → D′ is a K-quasiconformal homeomorphism.

List of properties of quasiconformal homeomorphisms:

1. f is differentiable almost everywhere in D.

2. Partial derivatives of f are locally in L_{2}(D).

3. Volume is absolutely continuous function under quasiconformal maps. Thus, the Jacobian determinant J_{f} is nonzero almost everywhere in D.

4. K(f) = K(f^{−1}), K(f ∘ g) ≤ K(f) · K(g) and K_{O}(f ∘ g) ≤ K_{O}(f) · K_{O}(g) for any pair of quasiconformal mappings f, g.

5. Suppose that f_{j} is a sequence of K_{j}-quasiconformal mappings of D into 𝕊^{n} so that :

(a) f_{j} fix three distinct points a_{1}, a_{2}, a_{3} ∈ D;

(b) K_{j} ≤ K < ∞.

Then f_{j} has a subsequence which is uniformly convergent to a

K-quasiconformal homeomorphism of D.

Now we assume that n = 2, D ⊂ \overline{ℂ} = 𝕊^{2}.

6. An orientation-preserving homeomorphism f : D → D′ is K-quasiconformal iff the following conditions are satisfied:

• f is absolutely continuous on almost every coordinate line in D;

• |∂f/∂z̄| ≤ k|∂f/∂z| where k = \frac{K−1}{K+1} < 1.

The measurable function

μ(z) = \frac{∂f/∂z̄}{∂f/∂z}

is called the complex characteristic (or Beltrami differential) of f. Orientation reversing quasiconformal homeomorphisms f : D → D′ are compositions of quasiconformal homeomorphisms and orientation reversing Moebius transformations.

7. Consider the Beltrami equation:

μ(z) = \frac{∂f/∂z̄}{∂f/∂z}

where μ ∈ L_{∞}(D) and ||μ|| ≤ k < 1. This equation has a solution f which is a quasiconformal homeomorphism of D, the mapping f is uniquely determined up to postcomposition with conformal mappings.

8. We normalize solutions of the Beltrami equation with the Beltrami differential μ by requiring solutions f to fix points ∞, a_{1} ≠ a_{2} ∈ ℂ. Then solutions f_{μ} of the Beltrami equation depend complex-analytically on μ.

9. Suppose that μ is a Beltrami differential in a domain D ⊂ \overline{ℂ}, h is a conformal automorphism of D. We define the action of h on μ by the formula

h_{∗}μ(z) = μ(hz)\overline{h′(z)}/h′(z)

For domains D let Conf(D) denote the group of conformal automorphisms of D. Suppose that Γ ⊂ Conf(D). Solutions f = f_{μ} of the Beltrami equation with the Beltrami differential μ conjugate Γ into the group of conformal automorphisms of f(D):

fΓf^{−1} ⊂ Conf(f(D))

if and only if the Beltrami differential μ is Γ-invariant.

5.2 Riemann surfaces

Recall that a Riemann surface is a 1-dimensional complex manifold. Each Riemannian metric on a 2-dimensional surface S is conformally flat, hence (if S is oriented) each Riemannian metric on S determines a complex structure on S. If S is a Riemann surface and ds^{2} is a Riemannian metric on S which determines the given complex structure on S, then ds^{2} is said to be compatible with S. A Riemann surface S is said to have finite type if it is conformally equivalent to a Riemann surface which is obtained from a compact Riemann surface S̄ by removing a finite number of points. The points removed from S̄ are called punctures on S. The genus of S̄ is called the genus of S. Let S be a Riemann surface of finite type of genus g with n punctures. If g = 0, n = 0 then S is said to have spherical conformal type, if g = 1, n ≤ 2 or (g, n) = (1, 0) then S is said to have Euclidean conformal type. In all other cases S has hyperbolic conformal type.

Theorem 5.1 (The uniformization theorem.) (1) If S is a Riemann surface of hyperbolic type or spherical type then S admits a complete metric of the curvature ±1 (negative in the hyperbolic case and positive in the spherical case) of finite area compatible with the complex structure on S. (2) If S has Euclidean type then S admits a complete flat metric compatible with the complex structure on S.

The most interesting case for us is when S has hyperbolic type. Then S is conformally isomorphic to ℍ^{2}/Γ, where ℍ^{2} is the hyperbolic plane and Γ is a subgroup of P SL(2, ℝ) acting freely properly discontinuously on ℍ^{2}. (Here we identify P SL(2, ℝ) with the group of orientation-preserving isometries of ℍ^{2}.)

Exercise 5.2 Let Γ ⊂ P SL(2, ℝ) be a subgroup acting freely properly discontinuously on ℍ^{2} so that ℍ^{2}/Γ has finite area. Then the Riemann surface S = ℍ^{2}/Γ has finite hyperbolic type. (Hint: first show that the thick part of S is compact, see Section 6.13.)

Suppose that X, Y are Riemann surfaces . A homeomorphism f : X → Y is called locally quasiconformal if it is quasiconformal in local conformal coordinates of X and Y. The coefficient of quasiconformality K(f) is the supremum of coefficients of quasiconformality of the corresponding quasiconformal homeomorphisms u_{j} ∘ f ∘ v_{k}^{−1} between domains of ℂ (where u_{j}, v_{k} are local conformal coordinates on X, Y). If K = K(f) < ∞ then f is called K-quasiconformal. Similarly one defines the Beltrami differential μ_{f} = μ(z)\overline{dz}/dz of f. The differential μ(z)\overline{dz}/dz is a measurable section of the bundle κ̄ ⊗ κ^{−1}, where κ = κ_{X} is the canonical bundle of X.

5.3 Geometry of quadratic differentials

Suppose that S̄ is a compact Riemann surface, P ⊂ S̄ is a finite subset disjoint from the boundary of S̄. Let ϕ = ϕ(z)dz^{2} be a nonzero (holomorphic) quadratic differential on S. We require ϕ to have at worst simple poles at the punctures p ∈ P. Let C := C(ϕ) denote the critical set of ϕ, i.e. finite collection of points on S where ϕ(z) = 0. Choose a base-point z_{0} ∈ S \ C. On the surface S \ C we have a multivalued locally-univalent holomorphic function

f : S \ C → ℂ, f(z) = ∫_{z_{0}}^{z} \sqrt{ϕ(z)}dz

This function is called the natural parameter associated with ϕ. We define a Riemannian metric on the surface S \ C by the formula:

ρ(z)|dz| := |ϕ(z)|^{1/2}|dz|

The function f(z) is a local isometry between this metric and the flat metric in ℂ. The metric ρ(z)|dz| extends to a path-metric d_{ρ} on the whole surface S. Near each critical point z_{j} ∈ C the total angle of d_{ρ} around z_{j} equals π(k_{j} + 2) > 2π, where k_{j} is the order of zero of ϕ(z) at z_{j}. Thus ρ(z)|dz| is a Euclidean metric with conical singularities > 2π on S. If P ≠ ∅ then the metric d_{ρ} on S is not complete (because ϕ has at worst simple poles at the punctures). However, if S is compact, the lift of d_{ρ} to the universal cover S̃ of S defines a CAT (0)-metric on S̃.

Besides the metric ρ(z)|dz|, the quadratic differential ϕ defines a pair of foliations of S − C by “horizontal” and “vertical” geodesics of the metric ρ(z)|dz|. They are called horizontal and vertical foliations of ϕ. Namely, pull-back the family of horizontal (resp. vertical) lines from ℂ to S − C via the map f and call them horizontal (resp. vertical ) trajectories of ϕ. Different branches of the map f define the same family of horizontal/vertical trajectories. If ∂S is nonempty we require:

For each component L ⊂ ∂S, L ∩ C ≠ ∅.

Near each critical point z_{j} ∈ C(ϕ) the vertical and horizontal foliations become singular: if the order of z_{j} equals k = k_{j}, then there are k + 2 horizontal (and the same number of vertical) trajectories of ϕ emanating from z_{j}. Besides nonsingular horizontal (vertical) trajectories of ϕ we will also use critical ones: those which contain critical points. Namely, consider a locally-injective continuous map

q : ℝ → S

so that on ℝ \ q^{−1}(C) this map is an isometry (with respect to the metric ρ) whose image consists of intervals of horizontal trajectories of ϕ. We call q a critical horizontal trajectory of ϕ. After being lifted to the universal cover S̃ is becomes a piecewise-smooth embedding of ℝ into S̃ which is a geodesic with respect to the metric d̃_{ρ}.

If S is compact then the group F = π_{1}(S) acts on (S̃, d̃_{ρ}) isometrically with compact quotient. Embed S̃ in ℍ^{2} as a convex domain with geodesic boundary.

Thus d̃_{ρ} is quasi-isometric to the hyperbolic metric, in particular lifts of trajectories of ϕ to ℍ^{2} are quasi-geodesics of the hyperbolic metric. We conclude that there is a canonical homeomorphism between the ideal boundary ∂_{∞}ℍ^{2} (with respect to the hyperbolic metric) and ∂_{∞}ℍ^{2} (with respect to the metric d̃_{ρ}).

Definition 5.3 Let f : ℝ → (S, ρ) be a local isometric map whose image lies in a single trajectory of ϕ. We assume that f can be approximated (uniformly on compacts) by locally isometric maps f_{t} whose images are nonsingular trajectories of ϕ. Then the map f is called a line in S. If L is the image of a lift of f to the universal cover S̃ of S then we will refer to L as a line as well.

Note that lines L ⊂ S̃ can be characterized as follows. Let ϕ̃ denote the lift of ϕ to S̃. Let S̃_{+}, S̃_{−} be the two connected components of S̃ − L. Then either S̃_{+} or S̃_{−} is disjoint from the trajectory of ϕ̃ which contains L. It is clear that if f_{n} : ℝ → L_{n} is a sequence of lines in S̃ which converge uniformly on compacts to a map f_{∞} then the image of f_{∞} is again a line.

5.4 Teichmüller spaces of Riemann surfaces

Suppose that S, S′ are connected Riemann surfaces of finite type, ψ : π_{1}(S) → π_{1}(S′) is an isomorphism. This isomorphism is said to preserve the peripheral structure of these surfaces if for any element γ ∈ π_{1}(S) corresponding to a puncture of S, its image ψ(γ) corresponds to a puncture of S′. If S, S′ are oriented then we call an isomorphism ψ : S → S′ admissible if the induced isomorphism H_{2}(S, ∂S; ℤ) → H_{2}(S′, ∂S′; ℤ) sends the fundamental class to the fundamental class and ψ preserves the peripheral structure. Suppose that S = ℍ^{2}/F, S′ = ℍ^{2}/F′ are hyperbolic surfaces where F, F′ ⊂ P SL(2, ℝ) are Fuchsian groups. An isomorphism ψ : F → F′ corresponds to an admissible isomorphism of the fundamental groups provided that it is type-preserving, i.e.: it sends parabolic elements to parabolic, hyperbolic elements to hyperbolic and respects the orientation^{2}.

Proposition 5.4 A conformal automorphism of a hyperbolic Riemann surface of finite area is homotopic to the identity iff it equals the identity.

Theorem 5.5 (Nielsen’s realization theorem). Any admissible isomorphism

ψ : π_{1}(S) → π_{1}(S′)

is induced by an orientation-preserving quasiconformal homeomorphism h : S → S′.

More generally:

Theorem 5.6 Suppose that F, F′ ⊂ Isom(ℍ^{2}) are Fuchsian groups (they could have torsion), such that ℍ^{2}/F, ℍ^{2}/F′ have finite area and

ψ : F → F′

is a type-preserving isomorphism. Then there exists a ψ-equivariant quasiconformal homeomorphism h : ℍ^{2} → ℍ^{2}.

I will give a sketch of the proof of this theorem in Section 9.3.

Fix once and for all a Riemann surface S of finite type, which has genus g and n punctures. Suppose that Y, Z are Riemann surfaces. Two quasiconformal homeomorphisms f : S → Y, g : S → Z are equivalent if f ∘ g^{−1} is homotopic to a conformal map between Z and Y. The equivalence class is called a marked Riemann surface and denoted [Y, f]. The space of marked Riemann surfaces [Y, f] is the Teichmüller space T (S) of the surface S. From now on we will consider Riemann surfaces of hyperbolic type (i.e. whose universal covers are conformal to the unit disc).

If S has hyperbolic type, then there is the following equivalent description of the Teichmuller space T (S). Let S = ℍ^{2}/F where F is a Fuchsian subgroup of P SL(2, ℝ). Consider the space Hom_{a}(F, P SL(2, ℝ)) of admissible isomorphisms between F and Fuchsian subgroups of P SL(2, ℝ). Take the quotient T_{R}(F) = Hom_{a}(F, P SL(2, ℝ))/P SL(2, ℝ) where P SL(2, ℝ) acts by conjugations. Nielsen’s theorem provides a natural bijection between T (S) and T_{R}(F). In this approach T (S) is the collection of marked hyperbolic surfaces of finite area which are homeomorphic to S (“marking” is given by a choice of an admissible isomorphism between the fundamental groups).

2ψ is induced by a map S → S′ which maps relative fundamental class to S to the relative fundamental class of S′.

Teichmüller metric on T (S). If p = [X, f_{0}], q = [Y, h_{0}] ∈ T (S) then put

d_{T} (p, q) = inf\{\frac{1}{2} log K(f ∘ h^{−1}) : (X, f) ∈ p, (Y, h) ∈ q\}

where K(τ) is the coefficient of quasiconformality of the quasiconformal homeomorphism τ between Riemann surfaces. This metric is complete and any pair of distinct points is connected by a unique geodesic. However d_{T} is not a Riemannian metric.

The infimum in the definition of Teichmüller distance is always achieved. The corresponding extremal quasiconformal homeomorphism τ = f ∘ h^{−1} is unique. Its Beltrami differential μ = μ_{τ} is described as follows. There exists a holomorphic quadratic differential φ on the surface Y (which is unique up to a positive scalar multiple), so that

μ = tφ̄/|φ|  (10)

where t > 0 is a constant. The extremal quasiconformal homeomorphism τ is real-analytic on Y everywhere except at a finite collection of points which is always nonempty provided that Y is compact: these singular points are zeroes of φ.

Remark 5.7 Existence of points where τ is not real-analytic fails if Y is not compact. As an example take two punctured tori T′, T′′ with flat metrics and an affine transformation A : T′ → T′′. This is a Teichmüller mapping.

Quasiconformal homeomorphisms τ : Y → X whose Beltrami differentials μ_{τ} are of the form (10) are called Teichmüller maps. The quadratic differential φ (which is defined up to a positive real multiple) is called the initial quadratic differential for τ. There is a unique (up to a positive real multiple) quadratic differential ψ on X so that:

• The mapping τ is affine with respect to Euclidean structures with conical singularities corresponding to φ and ψ;

• The mapping τ sends the horizontal foliation of φ to the horizontal foliation of ψ.

The quadratic differential ψ is called the terminal quadratic differential of τ.

Define the (Teichmüller) modular group (the other name is the Mapping class group) Mod_{S} as the quotient

Homeo_{+}(S)/Homeo_{0}(S)

where Homeo_{+}(S) is the group of orientation preserving self-homeomorphisms of S, we denote by Homeo_{0}(S) is the subgroup of homeomorphisms homotopic to id_{S}. Due to Nielsen’s Theorem 5.5, the group Mod_{S} acts on T (S) by the precomposition Mod_{S} ∋ g : [X, f] ↦ [X, f ∘ g]. The group Mod_{S} is isomorphic to

Out_{a}(π_{1}(S)) = Aut_{a}(π_{1}(S))/Inn(π_{1}(S))

where Aut_{a}(π_{1}(S)) consists of “admissible” automorphisms.

The space T (S) has a natural complex structure, it is biholomorphic to an open contractible domain in ℂ^{3g−3+n}. It is unknown if T (S) is holomorphically contractible, i.e. if there is a continuous family of holomorphic maps f_{t} : T (S) → T (S) so that f_{0} = id, f_{1} = const.

The Weil-Petersson metric on T (S). Besides the Teichmüller metric, the space T (S) has another useful metric d_{W P}, which is called the Weil-Petersson metric. The following are basic properties of this metric:

• d_{W P} is incomplete;

• (T (S), d_{W P}) is a Riemannian manifold;

• the manifold (T (S), d_{W P}) has negative sectional curvature;

• (T (S), d_{W P}) admits an exhaustion by compact convex subsets;

• the modular group Mod_{S} acts isometrically on (T (S), d_{W P}).

See [135], [208], [209], [210], [199], for definitions and proofs.

The group Mod_{S} acts on T (S) discretely, isometrically and biholomorphically. This action is not free, there is a subgroup of finite index Γ ⊂ Mod_{S} which acts freely. The quotient M(S) = T (S)/Mod_{S} is called the moduli space of complex structures on S. It can be viewed either as the space of unmarked complex structures of finite type on S or unmarked hyperbolic structures of finite area on S. Unless T (S) has complex dimension 1, the group Mod_{S} is the full group of isometries and the full group of holomorphic automorphisms of T (S).

The moduli space M(S) is non-compact. However, for given ϵ > 0 consider the subset M(S)_{ϵ} which consists of hyperbolic surfaces where the length of each homotopically nontrivial nonperipheral loop is at least ϵ.

Theorem 5.8 (Mumford compactness theorem.) The space M(S)_{ϵ} is compact.

The space M(S) has a natural compactification which is called the Mumford compactification. Namely, for the hyperbolic surface S define a partition P (S) as a decomposition of S by a finite collection of simple closed disjoint geodesics into open connected components. We treat these components as smooth surfaces and ignore the hyperbolic structure induced from S.

Define points Σ ∈ ∂_{m}(M(S)) to be complete hyperbolic metrics of finite area on various partitions P (S) (considered up to isometry). We topologize M̄(S) := M(S) ∪ ∂_{m}(M(S)) as follows. Pick a sufficiently small number μ > 0 (the Margulis constant for ℍ^{2}). The μ-thick part R_{[μ,∞)} of each point R ∈ M(S) consists of at most 2g + n − 2 components. (See Section 6.12 for more details about thick-thin decomposition.)

Then we say that a sequence Σ^{j} ∈ M(S) is convergent to a point Σ ∈ ∂_{m}(M(S)) if the following is satisfied:

• For each sequence x^{j} ∈ Σ_{[μ,∞)}^{n}, the sequence of pointed metric spaces (Σ^{j}, x^{j}) is convergent (in the quasi-isometric topology, see Section 4.9) to a component of Σ;

• Each component of Σ appears as such limit;

• Suppose that x_{1}^{j}, ..., x_{k}^{j} (k ≤ 2g + n − 2) are sequences of above points whose mutual distances in Σ^{j} tend to infinity and all the limits

lim_{j→∞}(Σ^{j}, x_{i}^{j}), 1 ≤ i ≤ k

are isometric. Then Σ contains at least k components isometric to this limit.

The union M̄(S) with this topology is compact and is called the Mumford compactification of M(S). Mumford proves that M̄(S) has a structure of projective variety which is consistent with the natural complex structure on M(S), see [155].

Geometrically, convergence of a sequence Σ^{j} to a point Σ ∈ ∂_{m}M(S) can be described as follows. There is a finite collection of simple closed disjoint geodesics γ_{1}^{j}, ..., γ_{k}^{j} on Σ^{j} whose length tends to zero as j → ∞. As j increases, these geodesics become more and more remote from the ϵ-thick part of the surfaces Σ^{j} and they become “pinched” in the limit. The limiting surface topologically is S − {γ_{1}, ..., γ_{k}}.

5.5 Locally finite subgroups of Mod_{S}

The following is the so called Nielsen realization problem which was was solved by S. Kerchkoff in [116], different solutions were found later in [210], [199] and combination of [66], [202], [37].

Theorem 5.9 Suppose that S is a complete hyperbolic surface of finite area. Then any finite subgroup F ⊂ Mod_{S} has a fixed-point [X, h] ∈ T (S). This means that F can be realized as a group of conformal automorphisms of a Riemann surface X.

Sketch of the proof: I will outline a proof following [210]. The difficult part of the proof is to establish the above properties of the Weil-Petersson metric on T (S); once it is done the proof is finished as follows. Take a point x ∈ T (S) and consider the orbit F · x = {x_{1}, ..., x_{n}}. Since (T (S), d_{W P}) admits an exhaustion by compact convex subsets, the convex hull of F · x is a compact convex subset Q ⊂ T (S). Clearly Q is a CAT (0)-space and F acts on Q isometrically. Thus, according to Proposition 4.2, F has a fixed point in Q ⊂ (T (S), d_{W P}). □

Corollary 5.10 Let S_{1}, ..., S_{k} be Riemann surfaces of finite type, T = T (S_{1}) × ... × T (S_{k}) is the product of their Teichmüller spaces. We give T the Riemannian metric of direct product of Weil-Petersson metrics on the factors T (S_{j}). Then any finite group of isometries F ⊂ Isom(T) has a fixed point on T.

Proof: Each space (T (S_{j}), d_{W P}) is a CAT (0)-space. Products of compact convex subsets in (T (S_{j}), d_{W P}) exhaust T and we use the fact that the direct product of CAT (0)-spaces is again a CAT (0)-space. □

We will use this corollary in Section 19.7.

I recall that a group G is called locally finite if any of its finitely-generated subgroups is finite. A group G is called a torsion group if every element of G has finite order. Note that any locally finite group is a torsion group. Let G = ℚ/ℤ where we consider both ℚ and ℤ as additive groups. Then the group G is locally finite. For a long time it was unknown if there are finitely generated infinite torsion groups (this is a version of the “Burnside Problem”), first such examples were constructed in [79]. It is however unknown if infinite torsion groups can be finitely presentable.

Corollary 5.11 (J. Gilman [75].) (1) Any locally-finite subgroup of Mod_{S} is finite. (2) Any torsion subgroup of Mod_{S} is finite.

Proof: There is an upper bound (due to Hurwitz) on the order of finite groups of conformal automorphisms of Riemann surface X of the fixed finite hyperbolic type:

|Aut(X)| ≤ 42(2g − 2 + n)

where g is the genus of X and n is the number of punctures. This estimate combined with Theorem 5.9 implies the first assertion.

There is also an alternative way to prove the first assertion. Namely, there is another classical theorem due to Hurwitz: if α is a conformal automorphism of X which acts trivially on H_{1}(X, ℤ), then α = 1. Consider the natural homomorphism

τ : Mod_{S} → GL(H_{1}(S, ℤ)) ≅ GL(2g + n − 1, ℤ) or ≅ GL(2g, ℤ)

Hurwitz theorem implies that for any torsion group G ⊂ Mod_{S} the homomorphism τ : G → GL(H_{1}(S, ℤ)) is a monomorphism. According to Shur’s Theorem, any finitely generated torsion subgroup in GL(n, ℂ) is finite (it also follows directly from Selberg Lemma, see Theorem 6.5). Selberg Lemma implies that there is an upper bound on orders of finite subgroups in GL(n, ℤ). Hence, the same upper bound holds for orders of torsion subgroups in Mod_{S}. □

6 Kleinian Groups

6.1 Representation varieties

Let Γ be a finitely-generated group, G be a semi-simple algebraic Lie group (the most interesting for us cases are: G = Isom(ℍ^{n}), G = P SL(2, ℂ), G = SL(2, ℂ)). Let G^{0} denote the connected component of the identity in G (with respect to the usual topology). Consider the quotient

R(Γ, G) = Hom(Γ, G)/G^{0}

where G^{0} acts on homomorphisms by conjugations: ρ ↦ ad_{g} ∘ ρ, where g ∈ G^{0} and ad is the adjoint action of G on itself. The topological space R(Γ, G) is not particularly nice, for instance in many interesting cases it is not Hausdorff. However it turns out that one can make sense of it as an algebraic variety. Namely, instead of taking the usual quotient of topological spaces one can take so called Mumford quotient, so that R(Γ, G) becomes a nice algebraic variety, called the character variety (see for instance [102, 125, 43, 151]). I will use the notation

R(Γ, G) = Hom(Γ, G)//G^{0}

for this quotient. Precise definition of of this quotient will not be essential here, the idea is to define first the ring of polynomial functions on the quotient Hom(Γ, G)//G^{0} as the subring of polynomial functions on the algebraic variety Hom(Γ, G) which are invariant under the action of G^{0}. From this ring one reconstructs the algebraic variety R(Γ, G).

Suppose that G = Isom(ℍ^{n}). A subgroup E of Isom(ℍ^{n}) is called elementary if it has either an invariant point in ℍ^{n} ∪ 𝕊_{∞}^{n−1} or an invariant geodesic L in ℍ^{n} (in the latter case E can change the orientation on L).

A representation ρ : Γ → G is said to be nonelementary if its image is a nonelementary subgroup in Isom(ℍ^{n}). Let Hom^{0}(Γ, G) denote the space of all nonelementary representations. It turns out that the spaces R(Γ, G) and R(Γ, G) are not much different, they are naturally homeomorphic on the subset Hom^{0}(Γ, G)/G^{0} of conjugacy classes of nonelementary representations. The variety R(Γ, G) need not to be a manifold. However if we restrict ourselves to the case: Γ = π_{1}(S), where S is a closed orientable hyperbolic surface of genus g and G is one of the groups: Isom(ℍ^{3}), P SL(2, ℂ), SL(2, ℂ), then Hom^{0}(Γ, G)/G^{0} is a smooth complex manifold of the complex dimension 6g − 6. This explains one of the reasons why we divide the space of homomorphisms by G^{0} rather than by G itself: in the case of G = SL(2, ℂ), P SL(2, ℂ) it is all the same, but if G = Isom(ℍ^{3}), then the group G does not act freely on Hom^{0}(Γ, G) and this action is not holomorphic (because of orientationreversing elements). If G = P SL(2, ℝ) then Hom^{0}(Γ, G)/G is a smooth manifold of the (real) dimension 6g − 6.

Similar results hold for relative parabolic representation varieties of the fundamental groups of punctured surfaces. First I will give a general definition. Suppose Γ is a finitelygenerated group and H := {H_{1}, ..., H_{q}} is a collection of its subgroups. Let G := Isom(ℍ^{n}). Consider the space of relative parabolic homomorphisms

Hom(Γ, H; G) = {ρ : Γ → G : ρ(γ) is parabolic for all γ ∈ H_{j}, j = 1, ..., q}

We define the relative parabolic representation variety as the quotient:

R(Γ, H; G) := [Hom(Γ, H; G) ∩ Hom^{0}(Γ, G)]/G^{0}

Then R(Γ, H; G) naturally sits in R(Γ, G) as an open subset of a subvariety. To simplify the terminology I will call R(Γ, H; G) simply a relative representation variety or parabolic representation variety. Now we again specialize our discussion to the case: S is a hyperbolic surface of genus g with q punctures. Let H_{j} be the cyclic groups generated by loops going around punctures of S. Then R(Γ, H; G) is a smooth complex manifold of the complex dimension 6g−6+2q if G = Isom(ℍ^{3}), P SL(2, ℂ), SL(2, ℂ); R(Γ, H; G) is a smooth manifold of the real dimension 6g − 6 + 2q if G = PSL(2, ℝ). See for instance [206].

If G = SL(2, ℂ) (and Γ is any finitely generated group), there is a particularly nice description of the variety R(Γ, SL(2, ℂ))y. Namely, for each element γ ∈ Γ we have the function

T r_{γ} : Hom(Γ, SL(2, ℂ)) → ℂ , T r_{γ}(ρ) = T race(ρ(γ))

(the character). It turns out that there is a finite collection of elements γ_{1}, ..., γ_{m} ∈ Γ so that the vector-function:

T : Hom^{0}(Γ, SL(2, ℂ))/SL(2, ℂ) → ℂ^{m}, T (ρ) = (T r_{γ_{1}}, ..., T r_{γ_{m}})

is injective and the affine variety equal to the closure of its image in ℂ^{m} is naturally identified with the Mumford quotient Hom(Γ, SL(2, ℂ))//SL(2, ℂ). See [43], [80] for an explicit construction of the elements γ_{j} and [151] for the case of representations to Isom(ℍ^{n}). As a corollary we have the following:

Proposition 6.1 There is a finite collection of elements α_{1}, ..., α_{k} ∈ Γ, so that the map

L : R(Γ, Isom(ℍ^{3})) → ℝ_{+}^{k}, L([ρ]) = (ℓ(ρ(α_{1})), ..., ℓ(ρ(α_{k})))

is proper. (I recall that ℓ(g) stands for the translational length of the element g ∈ Isom(ℍ^{3}).)

Proof: There is some difference between representations into Isom(ℍ^{3}) and SL(2, ℂ): not every representation to Isom(ℍ^{3}) has image in P SL(2, ℂ) and not every representation to P SL(2, ℂ) lifts to SL(2, ℂ). However after we take an index 4 subgroup in Γ this difference disappears and its clearly enough to check convergence on a finite-index subgroup. So we will work with representations to SL(2, ℂ).

Recall that the translational length of loxodromic elements g ∈ SL(2, ℂ) can be recovered from the trace t of g as:

t = λ + λ^{−1}, ℓ(g) = | log(|λ|)|

Thus, if ℓ(g) is bounded by a constant C, then 1 ≤ max(|λ|^{−1}, |λ|) ≤ e^{C} and therefore t belongs to a bounded domain in ℂ as well. If g is elliptic or parabolic then |T r(g)| ≤ 2 and there is nothing to check. □

Representations and flat bundles. There is a natural correspondence between representations and flat bundles. Suppose that M is a closed smooth manifold with the fundamental group Γ, G is a Lie group as above acting effectively on a space X. With any representation ρ : Γ → G we associate the flat X-bundle E_{ρ} = X ×_{ρ} M over M. We assume that ρ varies in a connected component C of Hom(Γ, G), then all the bundles E_{ρ} are smoothly isomorphic to a fixed bundle E. Thus we get a map ϕ̃ : C → F (E), where F (E) is the space of flat connections on the bundle E (provided with the C^{1}-topology). To make ϕ a bijection we have to divide C by the adjoint action of G^{0} (connected component of the identity) and the space F (E) by G_{0}, which is the group of gauge transformations homotopic to the identity. It is easy to verify that ϕ̃ projects to a bijection

ϕ : C/G^{0} → F (E)/G_{0}

which is an open map.

Theorem 6.2 ϕ is a homeomorphism.

Proof see in [78], [61]. (Actually C/G^{0}, F (E)/G_{0} have structure of analytic varieties and ϕ is an isomorphism of analytic varieties.)

6.2 Residual finiteness and Selberg lemma

Recall that a group G is called residually finite if there exists a chain of finite-index subgroups G_{i} ⊂ G such that

⋂_{i=1}^{∞} G_{i} = {1}

A group G is called Hopfian if any epimorphism f : G → G is injective.

Exercise 6.3 Show that every finitely generated residually finite group is Hopfian (see [128]).

Not every group is residually finite:

Exercise 6.4 Show that the Baumslag-Solitar group

⟨a, b|ab^{2}a^{−1} = b^{3}⟩

is not residually finite (see [12]). Hint: show that the homomorphism f : G → G, f(a) = a, f(b) = b^{2} is onto but is not injective.

However all finitely-generated subgroups of SL(n, ℂ) are residually-finite according to the following theorem known as “Selberg Lemma”:

Theorem 6.5 Suppose that G is a finitely-generated subgroup of SL(n, ℂ). Then G is residually finite (A. Malcev [128]) and there exists a finite-index subgroup G_{0} ⊂ G which is torsion-free (A. Selberg [183]).

One of corollaries of Thurston’s Hyperbolization Theorem is the following

Theorem 6.6 (J. Hempel, [90].) Fundamental group of every Haken manifold is residually finite.

There are other classes of groups which are known to be residually finite, for instance the Teichmüller modular group Mod_{S}. More generally, if G is a finitely generated residually finite group then the group Aut(G), which consists of automorphisms of G, is also residually finite, see [11].

6.3 Basics of discrete groups

In what follows we shall often use the fact that a closed subgroup of a Lie group is a Lie subgroup. A subgroup Γ of a Lie group G is called discrete if it is discrete in G as a topological subspace. In particular, a subgroup Γ is discrete iff cl_{G}(Γ) has dimension zero, i.e. the connected component of {1} in cl_{G}(Γ) is {1}.

Definition 6.7 Let Γ be a group of homeomorphisms of a topological space X. Then Γ acts properly discontinuously on X if for every compact K ⊂ X the intersections K ∩ hK is empty for all but finitely many h ∈ Γ.

Definition 6.8 Let Γ be as above and Γ_{0} be its subgroup. Then a subset S of X is called precisely invariant under Γ_{0} in Γ if h(S) = S for all h ∈ Γ_{0} and g(S) ∩ S = ∅ for all g ∈ Γ − Γ_{0}.

Definition 6.9 Let Γ ⊂ Isom(ℍ^{n}). Then x ∈ 𝕊_{∞}^{n−1} is a point of discontinuity for Γ if there is a neighborhood U of x such that U ∩ gU ≠ ∅ only for finitely many g ∈ Γ. Usually, this means that U ∩ gU ≠ ∅ for all g ∈ Γ \ {1}; the exceptional case is: x is a fixed point of a finite subgroup F ⊂ Γ. The domain of discontinuity Ω(Γ) consists of all points of discontinuity. It’s easy to see that this is an open subset of 𝕊_{∞}^{n−1}.

It follows from the convergence property of Moebius transformations (see Section 4.6) that Γ always acts properly discontinuously on Ω(Γ).

Definition 6.10 Discontinuous (the other name is Kleinian) group is a subgroup of Isom(ℍ^{n}) with nonempty domain of discontinuity.

Clearly any Kleinian group is discrete. The discreteness of Γ implies that it acts properly discontinuously (from the left) on the Lie group G = Isom(ℍ^{n}). However, ℍ^{n} = G/K where K is a maximal compact subgroup of Isom(ℍ^{n}) (which is conjugate to SO(n)). Therefore, compactness of K implies that any discrete group Γ acts properly discontinuously on ℍ^{n}.

Now we can define the limit set Λ(Γ) of a discrete group Γ ⊂ Isom(ℍ^{n}) as the set of accumulation points for the orbit Γx for some (any) x ∈ ℍ^{n}. If Γ is Kleinian then y ∈ Λ(Γ) iff for some (any) point x ∈ Ω(Γ) there is an infinite sequence of elements γ_{i} ∈ Γ such that lim_{i→∞} γ_{i}x = y.

Example 6.11 The subgroup P SL(2, ℤ) ⊂ P SL(2, ℂ) is Kleinian. The limit set of this subgroup is the extended real line 𝕊^{1} = ℝ ∪ {∞}. The group P SL(2, ℤ[i]) is discrete but is not Kleinian, its limit set is the whole sphere \overline{ℂ}.

More generally, a subgroup F ⊂ Isom(ℍ^{3}) will be called Fuchsian if it is finitely generated, the limit set of F is a round circle (or an extended straight line) and F preserves the orientation on Λ(F).

Remark 6.12 Classically such groups are called Fuchsian groups of the first kind.

The subgroup P SL(2, ℤ) ⊂ Isom(ℍ^{3}) is Fuchsian. Suppose that F is a Fuchsian group, h : \overline{ℂ} → \overline{ℂ} is a homeomorphism such that Q = hF h^{−1} ⊂ Isom(ℍ^{3}). Then Q is again a Kleinian group, the limit set of Q is the image of the round circle (the limit set of F) under the homeomorphism h. The group Q is called a quasifuchsian group.

Example 6.13 Suppose that we are given a collection of disjoint closed round disks

B_{1}, B_{2}, … B_{k}, B′_{1}, … B′_{k}

in ℂ and a family of Moebius transformations g_{j} : int(B_{j}) → ext(B′_{j}). Then the group Γ = ⟨g_{1}, …, g_{k}⟩ is a free Kleinian group of the rank k. The group G is called a classical Schottky group. If instead of round disks we take topological disks then the result is called a Schottky group. It is again free and discrete. The limit set of this group is a Cantor set (if k ≥ 2). It is obtained by removing from the complex plane the Γ-orbit of the domain

D = \overline{ℂ} − int(B_{1}) ∪ int(B_{2}) … ∪ int(B_{k}) ∪ int(B′_{1}) ∪ … ∪ int(B′_{k})

(See Figure 7.)

Figure 7: A Schottky group.

6.4 Elementary groups

An infinite discrete group Γ ⊂ Isom(ℍ^{n}) is elementary iff its limit set is either one or two points. In the latter case Γ contains an infinite cyclic subgroup Γ_{a} := ⟨γ⟩ of finite index, where γ is a loxodromic element.

The situation in the former case is more complicated. By conjugating the group Γ we get that Γ(∞) = ∞ where we identify 𝕊_{∞}^{n−1} with \overline{ℝ^{n−1}}. Thus Γ ⊂ Isom(ℝ^{n−1}). Hence by Bieberbach’s theorem, Γ contains a free Abelian subgroup Γ_{a} of finite index. The group Γ_{a} has rank r which is between 1 and n − 1. The number r will be called the virtual rank of Γ.

6.5 Properties of limit sets

Suppose that Γ is a discrete subgroup of Isom(ℍ^{n}).

• The limit set is either perfect (i.e. each point is an accumulation point) or consists of not more than 2 points. (In the latter case the group is elementary.)

• The limit set and discontinuity domain are Γ-invariant.

• Λ(Γ) = 𝕊_{∞}^{n−1} − Ω(Γ).

• If Γ is nonelementary then the loxodromic fixed points are dense in Λ(Γ).

The convex hull in ℍ^{n} of the limit set Λ = Λ(Γ) of a discrete group Γ ⊂ Isom(ℍ^{n}) will be denoted by C(Λ), it is called the Nielsen’s convex hull of Γ. We will be mainly interested in the case n = 3, when the boundary of the convex hull C(Λ) is a union of disjoint complete geodesics in ℍ^{3} and 2-dimensional hyperbolic polygons with vertices at \overline{ℂ} (these polygons could have infinitely many vertices).

Example 6.14 Let Λ ⊂ ℂ be an ellipse which is not a circle. Then the boundary of C(Λ) has two components each of which is foliated by complete geodesics.

Suppose that Γ ⊂ Isom(ℍ^{3}) is a nonelementary discrete group and G ⊂ Isom(ℍ^{3}) contains Γ as a normal subgroup. Then G must be also discrete. Indeed, by continuity, the group Γ is normal in the closure cl(G) of G in Isom(ℍ^{3}). The action by conjugations of the identity component cl(G)^{0} on Γ is trivial. Therefore cl(G)^{0} fixes pointwise the limit set of Γ. However, since Λ(Γ) contains more than 2 points, the group cl(G)^{0} must be trivial. Therefore G is discrete, otherwise cl(G) is a Lie subgroup and cl(G)^{0} ≠ {1}.

6.6 Quotient spaces of discrete groups

For each discrete group Γ ⊂ Isom(ℍ^{3}) we define the quotient-spaces:

M(Γ) = ℍ^{3}/Γ  (11)
CM(Γ) = C(Λ(Γ))/Γ  (12)

If Γ is torsion-free, then M(Γ), C(Γ) are hyperbolic manifolds, if it has torsion, then these spaces have more complicated orbifold structure which we shall discuss later. We shall mainly consider the case of torsion-free groups. The space CM(Γ) is called the convex core of M = M(Γ). If Γ is torsion-free, then the convex core may be defined as the minimal closed convex subset of M whose embedding induces a homotopy-equivalence to M.

Suppose that Ω(Γ) ≠ ∅. Then we can attach to M(Γ) the ideal boundary:

\overline{M}(Γ) = M(Γ) ∪ Ω(Γ)/Γ  (13)

Example 6.15 Let G be a Schottky group of the rank r. Then \overline{M}(G) is the handlebody of the genus r. Let F ⊂ P SL(2, ℝ) be a Fuchsian group. Then \overline{M}(F) = I × (ℍ^{2}/F).

6.7 Fundamental domains

Suppose that a group Γ acts properly discontinuously on a topological space X. Then a subset F ⊂ X is called a fundamental set for Γ if:

(i) the orbit ΓF is equal to X,

(ii) for each γ ∈ Γ we have γF ∩ F ≠ ∅ unless γ does not act freely on X, in which case we require γF ∩ F = F ix_{X}(γ) ∩ F.

The concept of a fundamental domain is more subtle. Let X be either ℍ^{3}, or ℍ^{3} ∪ Ω(Γ) or Ω(Γ). By a polygon in Ω(G) we shall mean a polygon bounded by circular arcs.

Definition 6.16 A fundamental domain (polyhedron) D for the action of a discrete group G on X is a codimension zero piecewise-smooth submanifold (subpolyhedron) of X such that:

(i)

⋃_{g∈Γ} g(cl_{X}D) = X  (14)
g(intD) ∩ intD = ∅, for all g ∈ Γ − {1}  (15)

(ii) the boundary of D in X is a piecewise-smooth (polyhedral) submanifold in X and it is divided in a union of smooth submanifolds (convex polygons) , which are called faces; for each face S there another face T and an element g = g_{ST} ∈ Γ − 1 so that gS = T (g is called a face-pairing transformation); g_{ST} = g_{T S}^{−1};

(iii) the orbit ΓD is locally finite in X, i.e. each compact in X intersects only finitely many members of the family {γD : γ ∈ Γ}.

Remark 6.17 Faces of D are not necessarily maximal smooth (totally geodesic) submanifolds of ∂D. If X = Ω(Γ), then by abusing notations we shall say that D is a fundamental domain for the action of Γ on \overline{ℂ}.

It follows that the quotient cl_{X}D/Γ is homeomorphic to X/Γ. It’s easy to see that the face-pairing transformations generate the group Γ. One can also “read off” the defining relations of Γ looking at the fundamental domain.

Theorem 6.18 Every discrete subgroup Γ ⊂ Isom(ℍ^{3}) has a fundamental domain.

Proof: I will give a proof in the case when Γ acts freely on X, otherwise one has to take into account the orbifold structure of X/Γ. The quotient X/Γ is a smooth manifold, hence it admits a triangulation T. Take a 1-dimensional graph L in X/Γ which is dual to this triangulation (i.e. vertices of L are centers of top-dimensional simplices of T and two vertices are connected by an edge in L if the corresponding simplices share a common face of codimension 1). Let R be a maximal subtree in L. Remove from T all closed codimension 1 simplices which are disjoint from R. The rest is simply-connected, so it has a homeomorphic lift D into X. It’s easy to see that cl_{X}(D) is a fundamental domain. □

There are some other ways to produce fundamental domains.

Definition 6.19 Let O ∈ ℍ^{3} be a point which is not a fixed point for any nontrivial element of the discrete group Γ. Define

D(O, Γ) = {x ∈ ℍ^{3} : d(O, x) ≤ d(x, gO), for all g ∈ Γ − {1}}  (16)

Then D(O, Γ) is a fundamental domain for the action of Γ in ℍ^{3} which is called the Dirichlet fundamental polyhedron for the group Γ with the center at O. See [13] or [133].

Note that each Dirichlet fundamental polyhedron is convex.

Examples of fundamental domains. The easiest example is a Schottky group. The complement to the union of disks which we used to define this group, is a fundamental domain for its action in \overline{ℂ}.

6.8 Poincare theorem on fundamental polyhedra

In this section we will give a general theorem which describes conditions under which a convex polyhedron in ℍ^{3} is a fundamental domain. This theorem is originally due to Poincare (who proved it for the hyperbolic plane), later (more general) versions are due to Alexandrov [6], Maskit [133], etc. We will present here a version valid in the hyperbolic 3-space ℍ^{3} following [133].

Start with a convex polyhedron P ⊂ ℍ^{3}. As in Section 4.7 this polyhedron is the complement to a countable collection of hyperbolic half-spaces F̂_{j}^{−} which form a locallyfinite family in ℍ^{3} (i.e. any compact intersects only finitely many of them). The hyperplanes F̂_{j} bounding these half-spaces define geometric faces of P. We define faces of P to be convex domains in geometric faces of P so that each geometric face contains at most two faces. Edges of P are 1-dimensional intersections between faces. Vertices are defined to be 0dimensional intersections of faces.

We assume that P satisfies the following axioms:

Axiom 1. For each face S there another face R (“paired with S”) and an element g = g_{SR} ∈ Γ − {1} so that gS = R, g is the “face-pairing transformation”; g_{SR} = g_{RS}^{−1} (as in the definition of Fundamental Domain).

Axiom 2. g_{SR}int(P) ∩ int(P) = ∅.

Example 6.20 Suppose that P itself is a hyperbolic half-space bounded by a hyperplane H, which is the only geometric face of P. Pick a geodesic e ⊂ H, it separates H into two faces. Now let g ∈ P SL(2, ℂ) be the rotation of order 2 around e: it necessarily pairs these faces.

If paired faces S, R are equal, then the Axiom 2 implies that g_{SR}^{2} = 1, which is called the reflection relation since necessarily g_{SR} is the reflection in S.

Face-pairing transformations induce an equivalence relation on P generated by: g_{SR}(x) ∼ x. Note that each interior point of P is equivalent only to itself. Let P^{∗} be the quotient of P by ∼, which is provided with the quotient topology, p : P → P^{∗} is the natural projection.

Axiom 3. p^{−1}(y) consists only of finitely many points for each y ∈ P^{∗}.

Next we define cycles corresponding to P. Suppose that e_{1} is an edge of P. Then e_{1} is adjacent to a face F_{1} of P. Take the face-pairing transformation g_{1} := g_{F_{1},F′_{1}} : F_{1} → F′_{1} and e_{2} = g_{1}(e_{1}). Then we proceed as follows: take F_{2} ≠ F′_{1} be the face adjacent to e_{2}, find a face-pairing transformation g_{2} := g_{F_{2},F′_{2}} : F_{2} → F′_{2} and continue until we return back to the original face-edge pair: (F_{1}, e_{1}) = (F′_{n}, e_{n+1}). The ordered collection of edges (e_{1}, e_{2}, ...e_{n−1}) is called an edge cycle, the number n is called the period of this cycle. Note that we allow e_{1} = e_{k} for some 1 < k < n in this cycle. The composition

T_{e_{1}} := g_{n} ∘ ... ∘ g_{1}

is called the cycle transformation at e_{1}. Clearly T_{e_{1}} keeps e_{1} and the face F_{1} invariant. However apriori it could be a hyperbolic translation along e_{1}.

Axiom 4. We require T_{e} = 1 for each edge e. The space P^{∗} has a natural path metric d^{∗} induced from ℍ^{3}. Our last axiom is:

Axiom 5. The metric space (P^{∗}, d^{∗}) is complete.

Remark 6.21 This axiom is clearly satisfied by all compact polyhedra P.

Note that all but the last axiom are local in nature, to verify them we need to look only at each individual face and edge. To the contrary, the Axiom 5 requires working with P^{∗} as a global object. Below we shall describe some conditions which guarantee that the Axiom 5 holds.

Theorem 6.22 Suppose that P satisfies the Axioms 1 through 5. Then P is a fundamental polyhedron for the group G generated by face-pairing transformations. The group G has the presentation:

⟨g_{S,R} | g_{SS}^{2} = 1, T_{e} = 1⟩

where S, R, e ran through the list of faces and edges of P.

For proof see [133].

Suppose that P satisfies the Axioms 1 through 5, let P̄ be the compactification of P in \overline{ℍ^{3}} = ℍ^{3}∪𝕊_{∞}^{2}, P_{∞} is the interior of the subset ∂_{∞}P := P̄∩𝕊_{∞}^{2}. Then it is easy to derive from Theorem 6.22 and definition of fundamental polyhedra that P_{∞} is a fundamental domain for the action of G on \overline{ℂ} = 𝕊_{∞}^{2}.

Particular cases:

Reflection groups. Suppose that each face of P is paired to itself only (each facepairing transformation is a reflection in that face), dihedral angles of P between adjacent faces F_{i}, F_{j} are π/n_{ij} where n_{ij} > 1 are integers. If faces F_{i}, F_{j} are not adjacent we put n_{ij} = 0. Then P satisfies the Axioms 1 through 5 and the group G generated by reflections g_{j} := g_{S_{j}S_{j}} has the following presentation:

⟨g_{j}, j = 1, 2, ...|(g_{j})^{2} = 1, (g_{i}g_{j})^{n_{ij}} = 1⟩

Indeed, the only tricky condition is the Axiom 5, however in our case P^{∗} = P is obviously complete as a closed subset of ℍ^{3}.

Consider the following example: take three round disks D_{1}, D_{2}, D_{3} in \overline{ℂ} such that the exterior angles of their intersection are: π/n, π/m, π/q, where n, m, q = 2, 3, 4, ...∞. The complement of their union is a pair of triangles T, T′ ⊂ \overline{ℂ} bounded by arcs of circles (if n = m = q = ∞ then these are ideal triangles). Let G be the group generated by reflections in the circles ∂D_{1}, ∂D_{2}, ∂D_{3}. This group is called the Schwarz triangle group T (n, m, q), its fundamental domain in \overline{ℂ} is P_{∞} := T ∪ T′. The convex hull of P_{∞} is a fundamental domain P of G in ℍ^{3}.

From now on we restrict ourselves to polyhedra with finitely many faces. Suppose that x_{1} ∈ ∂_{∞}P is a point of tangency between two faces F_{n}, F of P. Then we form a cycle corresponding to x similarly to the construction of edge cycles of P:

Take a side-pairing transformation g_{1} := g_{F_{1},F′_{1}} : F_{1} → F′_{1} and x_{2} = g_{1}(x_{1}). If x_{2} is not a point of tangency between two faces, we stop. Otherwise, let F_{2} ≠ F′_{1} be the face tangent to F′_{1} at e_{2}, find a face-pairing transformation g_{2} := g_{F_{2},F′_{2}} : F_{2} → F′_{2} and continue. We stop if either x_{n+1} is not a point of tangency or if x_{1} = x_{n+1} = g_{n}x_{n}. If the latter happens we call

T_{x_{1}} := g_{n} ∘ ... ∘ g_{1}

infinite cycle transformation at x_{1}. Clearly T_{x_{1}}(x_{1}) = x_{1}, apriori it could be a loxodromic element.

For instance, let P be bounded by two hyperplanes H_{1}, H_{2} which are tangent at a point x ∈ 𝕊_{∞}^{2} and g_{12} : H_{1} → H_{2} be the face-pairing transformation. Then

T_{x} = g_{12}

Axiom 5′. All infinite cycle transformations are parabolic.

Theorem 6.23 (B. Maskit [133]) If the Axioms 1 through 4 are satisfied (and P has only finitely many faces), then the Axiom 5 is equivalent to the Axiom 5′.

To construct a concrete example take a configuration of four unit round disks D_{1}, D_{2}, D_{3} and D_{4} ⊂ ℂ so that:

• These disks are orthogonal to a common circle C in ℂ.

• The disks D_{i}, D_{i+1} are tangent at points x_{ii+1} (i = 1, ..., 4 (mod 5)).

• The configuration of disks admits a symmetry of order 4 which cyclicly permutes these disks.

Now choose two Moebius transformations

g_{12} : D_{1} → (\overline{ℂ} − int(D_{2})), g_{34} : D_{3} → (\overline{ℂ} − int(D_{4}))

as follows: let L be straight line tangent to D_{1}, D_{2} at x_{12} and to D_{3}, D_{4} at x_{34}. Let g_{12} be composition of the symmetry in L followed by the inversion in ∂D_{2}, similarly let g_{34} be the composition of the symmetry in L followed by the inversion in ∂D_{4}. Let P_{∞} be the complement to the union of interiors of D_{1}, ..., D_{4}, P be its convex hull in ℍ^{3}.

Then the Axiom 5′ is satisfied for the polyhedron P and the group G generated by the transformations g_{12}, g_{34}.

6.9 Geometrically finite groups

A discrete subgroup Γ ⊂ Isom(ℍ^{n}) is called a lattice if ℍ^{n}/Γ has finite volume. This lattice is called uniform if the quotient is compact. Examples of lattices are given by arithmetic groups. Instead of the general definition I will give several examples of arithmetic groups.

(1) The subgroup SL(2, ℤ), which consists of matrices with integer coefficients, is an arithmetic subgroup of SL(2, ℝ).

(2) Recall that Gaussian integers ℤ[i] are complex numbers where both real and imaginary parts belong to ℤ. The subgroup SL(2, ℤ[i]), which consists of matrices with Gaussian integer coefficients, is an arithmetic subgroup of SL(2, ℂ).

The groups SL(2, ℤ) and SL(2, ℤ[i]) act as nonuniform lattices on ℍ^{2} and ℍ^{3} respectively.

(3) Consider the quadratic form φ = x_{1}^{2} + … x_{n}^{2} − \sqrt{p}x_{n+1}^{2}, where p is a prime number. The group of linear automorphisms Aut(φ) of φ is isomorphic to SO(n, 1). Now, let Γ be the group of matrices SL(n + 1, ℤ) ∩ Aut(φ). One can show that Γ is a uniform lattice in Aut(φ).

The concept of a lattice is generalized by the notion of a geometrically finite group.

Definition 6.24 Suppose that Γ ⊂ Isom(ℍ^{3}) is a discrete group. Then Γ is called geometrically finite if for some (any) ϵ > 0 the ϵ-neighborhood Nbd_{ϵ}(CM(Γ)) of the convex core in M(Γ) has finite volume.

As we shall see later there are several alternative descriptions of geometrically finite groups, see Section 6.13.

Definition 6.25 A discrete group Γ ⊂ Isom(ℍ^{3}) is called geometrically finite if it has a convex fundamental polyhedron P ⊂ ℍ^{3} with finitely many faces.

Remark 6.26 Here by “face” we mean a subset of ∂P which appears in the definition of a fundamental domain.

Thus, each group satisfying the Definition 6.25 is finitely presentable.

Definition 6.27 A discrete group Γ ⊂ Isom(ℍ^{3}) is geometrically finite if its Dirichlet fundamental polyhedron D(O, Γ) ⊂ ℍ^{3} has finitely many faces.

To give other descriptions of geometrically finite groups we need few more definitions. Let Γ ⊂ Isom(ℍ^{3}) be a discrete nonelementary group.

Theorem 6.28 (A. Beardon and B. Maskit [14], see also the paper of B. Bowditch [29].) The following conditions are equivalent for each z ∈ Λ(Γ):

(1) There exists a sequence \{g_{n} ∈ Γ\} such that for each x ∈ \overline{ℂ} − \{z\} the sequence \{(g_{n}x, g_{n}z)\} is relatively compact in \overline{ℂ} × \overline{ℂ} − Diagonal;

(2) There exists a sequence \{g_{n} ∈ Γ\} such that for some subset W ⊂ \overline{ℂ} − \{z\}, which consists of at least two points, and for every point x ∈ W the sequence {(g_{n}x, g_{n}z)} is relatively compact in \overline{ℂ} × \overline{ℂ} − Diagonal;

(2’) There exists a sequence {g_{n} ∈ Γ} such that for every point x ∈ Λ(Γ) − {z} the sequence {(g_{n}x, g_{n}z)} is relatively compact in Λ(Γ) × Λ(Γ) − Diagonal;

(3) For each (for some ) point O ∈ ℍ^{3} there exists a sequence {g_{n} ∈ Γ}, a number r < ∞ and a geodesic σ emanating from z such that g_{n}^{−1}(O) converges to z inside the tubular neighborhood U_{r} = Nbd_{r}(σ). (I.e. lim_{n}^{c} g_{n}^{−1}(O) = z.)

Proof: The implications (1) ⇒ (2), (1) ⇒ (2′) are obvious. Let us prove that (2 ⇒ 3). Let σ be any geodesic as in (3) which is asymptotic to the points z and y ∈ W. Then the sequence of geodesics g_{n}σ intersects certain ball B(O, r) in ℍ^{3}. Thus, d(g_{n}^{−1}O, σ) ≤ r. The sequence g_{n}^{−1}O cannot accumulate at any point of ℍ^{3}, suppose that a subsequence g_{n_{k}}^{−1}O = O_{k} accumulates at y. Then we can take another geodesic σ′ emanating from z and asymptotic to y′ ∈ W − {y}. Then again we must have: O_{k} accumulates to y′ or z, which contradicts the assumption that it accumulates to y. This contradiction proves that O_{n} converges to z.

The proof of the implication (2′ ⇒ 3) is similar.

Now let us prove (3 ⇒ 1). Take \overline{ℂ} ∋ x ≠ z and a geodesic σ′ asymptotic to z and x. Since σ, σ′ are asymptotic at z we conclude that O_{n} converges to z within the 2r-neighborhood σ′. This implies that geodesics g_{n}(σ′) intersect B(O, r) for all sufficiently large n. The property (1) follows. □

Definition 6.29 A point p ∈ Λ(Γ) is called a conical limit point (the other name is a point of approximation) if one of the equivalent conditions in Theorem 6.28 is satisfied.

Example 6.30 Let p be a fixed point of a loxodromic element g ∈ Γ. Then p is a conical limit point. In the contrast, parabolic fixed points are never conical limit points.

Remark 6.31 In the Property (2) of Theorem 6.28 it is necessary to assume that W contains at least two points, otherwise (2) does not imply (3).

Another way to describe conical limit points is to look at the quotient. Let l = l_{z} ⊂ ℍ^{3} be a geodesic ray asymptotic to z ∈ Λ(Γ). Let ℓ = ℓ_{z} be the projection of l to the manifold M(Γ). Then (3) is equivalent to the property:

(4) there exists a compact subset C ⊂ M(Γ) such that the ray ℓ intersects C “infinitely many times” (i.e. there exists a sequence u_{n} ∈ ℓ such that distance between u_{o} and u_{n} on ℓ tends to infinity but their distance in M(Γ) is bounded). Such rays in M(Γ) are called recurrent.

WARNING. Recurrence of ℓ does not mean that the ray ℓ is relatively compact in M(Γ). For example, suppose that Γ is a nonuniform lattice in ℍ^{3} (i.e. the quotient ℍ^{3}/Γ is not compact). Then almost all points of 𝕊^{2} = \overline{ℂ} are points of approximation but for almost all points z ∈ \overline{ℂ} the rays ℓ_{z} are not relatively compact (it follows from ergodicity of the geodesic flow of M(Γ)).

Definition 6.32 A point p ∈ Λ(Γ) is called a cusped parabolic point if its stabilizer in Γ is an almost Abelian group A containing parabolic elements such that Λ(Γ)/A is compact.

In particular, if the virtual rank of A is 2 then the point p is always a cusped parabolic point. Another name for such points is bounded parabolic fixed points, see [29].

Definition 6.33 A discrete subgroup Γ ⊂ Isom(ℍ^{n}) is geometrically finite if every limit point of Γ is either a point of approximation or is a cusped parabolic point.

Note that both points of approximation and cusped parabolic points are defined purely topologically in terms dynamics of Γ on its limit set. (Use the Condition (2’) in Theorem 6.28 to define the points of approximation.) In particular, if Γ, Γ′ are discrete subgroups of Isom(ℍ^{n}), the group Γ is geometrically finite and f : Λ(Γ) → Λ(Γ′) is a homeomorphism which conjugates the action of Γ to the action of Γ′, then Γ′ is also geometrically finite.

6.10 Kleinian groups and Riemann surfaces

In what follows we shall need some facts about conformal geometry of Riemann surfaces. Suppose that R is the rectangle with the hight h and width w. Then the conformal modulus of R is defined as h/w. Suppose that A is a Riemannian product A = [−b, b] × 𝕊^{1}, where 0 < b ≤ ∞, and 𝕊^{1} = 𝕊_{ℓ}^{1} is the circle of the length ℓ > 0. Define the conformal modulus μ(A) to be the ratio 2|b|/ℓ. If M is a Riemann surface homeomorphic to A then, according to the uniformization theorem, M is conformally equivalent to a cylinder A′ with the product metric. Then μ(M) := μ(A′). It is easy to see that the modulus is a conformal invariant.

Let S be a Riemann surface and γ be a closed loop on S whose free homotopy class will bel denoted [γ]. Fix a metric ds in the conformal class of S, denote the area element of this metric by dvol. Suppose that ρ is a nonnegative Borel measurable function on S, we equip S with a (measurable) metric ρds and define the ρ-length of γ as

L_{ρ}(γ) = inf_{α∈[γ]} ∫_{α} ρds  (17)

For the metric ρds define the area A_{ρ}(S) as the integral

A_{ρ}(S) = ∫_{S} ρ^{2}dvol  (18)

Define the extremal length E_{S}(γ) of the curve γ on S as

E_{S}(γ) := sup_{ρ}{L_{ρ}(γ)^{2}/A_{ρ}(S) : 0 < A_{ρ}(S) < ∞}  (19)

The extremal length E_{S}(γ) depends only on the conformal structure of S and on the homotopy class of γ. It is easy to see from the definition of extremal length that if S is conformally embedded in a Riemann surface S then E_{S}(γ) ≤ E_{M}(γ) (just take an extremal metric ρ on S and extend it by zero to the rest of the surface M).

Lemma 6.34 (See [4].) Choose the product metric on the annulus A = 𝕊_{ℓ}^{1} × [−b, b], let [γ] be the generator of π_{1}(A). Then this flat metric is extremal in the definition of the extremal length of γ and the extremal representative α of γ is any horizontal loop. If S is a flat torus and γ ⊂ S is simple, homotopically nontrivial loop then the extremal length E_{S}(γ) is equal to the length of a geodesic representative of γ in S.

This lemma implies that the conformal modulus of a ring is superadditive in the following sense. Let A be an annulus, γ ⊂ int(A) be a simple homotopically nontrivial loop which separates A into two annuli A′, A′′. Then μ(A) ≥ μ(A′) + μ(A′′).

Now suppose that γ ⊂ S is a simple nontrivial loop. We define the conformal modulus mod(γ) to be the supremum of conformal moduli of all cylinders which are embedded in S and homotopic to γ. The following theorem establishes relation between the extremal length and the modulus:

Theorem 6.35 (See [4] .)

mod(γ)^{−1} = E_{S}(γ)  (20)

Example 6.36 Let h ∈ P SL(2, ℝ) be a hyperbolic element with the translational length ℓ(h). Then the modulus of the cylinder A = ℍ^{2}/⟨h⟩ is equal to π/ℓ(h).

Proof: Using the exponential map we represent A as the Euclidean rectangle of the shape π × ℓ(h) with two sides of the length π identified. Then

μ(A) = [ℓ(h)^{2}/(πℓ(h))]^{−1} = π/ℓ(h)

□

Suppose that Γ ⊂ P SL(2, ℂ) is a nonelementary Kleinian group and D ⊂ Ω(Γ) is a simply–connected component with the stabilizer Γ. Then D is conformally–equivalent to the upper half-plane ℍ^{2}. Therefore S = D/Γ has to be a Riemann surface of hyperbolic conformal type. Let h denote the hyperbolic metric on S induced by the conformal map between D and ℍ^{2}. Suppose that g ∈ Γ is an element of infinite order representing a geodesic loop α on S of the length ℓ(α) (with respect to the metric h). Recall that ℓ_{ℍ^{3}}(g) denotes the translational length of the isometry g acting on ℍ^{3}.

Theorem 6.37 (L. Bers, B. Maskit)

ℓ_{ℍ^{3}}(g) ≤ 2ℓ(α)  (21)

Proof: First note that the assertion is true if g is parabolic. Consider the torus (or annulus in the parabolic case)

T = Ω(⟨g⟩)/⟨g⟩

which contains the annulus A = D/⟨g⟩. Therefore mod_{T} (α) ≥ mod_{A}(α) = π/ℓ(α). If α is homotopic to a puncture on S then mod_{T} (α) = ∞ and thus T is biholomorphic to ℂ^{∗} which means that g is parabolic. Consider the loxodromic case. Then using the exponential map we represent T as the parallelogram P with the base 2π and of the height ℓ(g) with identified sides. Thus mod_{T} (α) = mod(T − α) and T − α is the annulus of the modulus at most 2π/ℓ(g). Whence

π/ℓ(α) ≤ 2π/ℓ(g)  (22)

□

Corollary 6.38 Suppose that in Theorem 6.37 the loop γ is homotopic to a puncture. Then ℓ(g) = 0, i.e. g is parabolic.

6.11 Kazhdan-Margulis-Zassenhaus lemma

In this section we discuss several results which are fundamental for studying discrete groups acting on symmetric spaces. The first one was proven by H. Zassenhaus in 1938, the second by D. Kazhdan and G. Margulis in 1968. Both results remained unnoticed^{3} to the people working the the theory of Kleinian groups until Thurtson came to the field in 1970-s.

Theorem 6.39 (H. Zassenhaus, [211]) Let G be a Lie group, then there is a neighborhood of identity U_{Z} ⊂ G such that for each discrete subgroup Γ ⊂ G the group generated by Γ ∩ U_{Z} is nilpotent.

Proof: Let us compute the derivative of the commutator map [·, ·] : G × G → G at the identity; it is enough to consider the case when G is linear since the center of G does not matter in this computation. Let η, ζ be elements of the Lie algebra of G, t, s ∈ ℝ. Then

[1 + tη, 1 + sζ] = 1 + ts(ηζ − ζη) + higher order terms

3Instead, various replacements of these results were proven by Shimizu, Keen, Jorgensen and others via methods specific to SL(2, ℂ).

which implies that the derivative of [·, ·] at (1, 1) is identically zero. Hence the map [·, ·] is a strict contraction with respect to the both variables in a neighborhood U of the identity. Thus, for any elements x_{1}, ..., x_{m} ∈ Γ ∩ U the iterated commutarors

y_{n} := [… [x_{1}, x_{2}], …, x_{n}]

belong to U and morevover lim_{n} y_{n} = 1. Since the group Γ is discrete, any such sequence must eventually terminate at the identity. Therefore there is a number N such that for any choice of x_{1}, ..., x_{N} ∈ Γ ∩ U the iterated commutaror y_{N} equals 1. We claim that this implies that the group Γ is nilpotent. The proof is the reverse induction on n. Let Γ_{n} be the subgroup of Γ generated by n-fold commutators y_{n}. For any generator x_{i} of Γ and any y_{n} ∈ Γ_{n} we have: [y_{n}, x_{i}] = y_{n}x_{i}y_{n}^{−1}x_{i}^{−1} ∈ Γ_{n+1}, hence x_{i}y_{n}^{−1}x_{i}^{−1} ∈ Γ_{n} which implies that Γ_{n} is a normal subgroup of Γ. The subgroup Γ_{N} is trivial, hence it is in the center of Γ. To show that Γ_{n−1}/Γ_{n} is in the center of Γ/Γ_{n} it is enough to verify that for elements z and w of generating sets of Γ_{n−1}/Γ_{n} and Γ/Γ_{n} the commutator [z, w] is trivial. The group Γ is generated by x_{1}, ..., x_{m} ∈ Γ∩U, the group Γ_{n−1} is generated by n−1-fold commutators y_{n−1} of the above elements x_{i}. Thus Γ_{n−1}/Γ_{n} and Γ/Γ_{n} are generated by projections x̄_{i}, ȳ_{n−1} of the elements x_{i}, y_{n−1}. By definition of Γ_{n} we have: [y_{n−1}, x_{i}] ∈ Γ_{n}, thus [ȳ_{n−1}, x̄_{i}] = 1. □

The neighborhood U = U_{Z} is called Zassenhaus neighborhood .

Let X = G/K be the homogeneous space associated to G, where K is a maximal compact subgroup of G. We assume that G is given a left-invariant Riemannian metric which is also right-invariant under K. Then X becomes a Riemannian manifold and G acts (on the left) by isometries of X.

Theorem 6.40 (D. Kazhdan and G. Margulis, [115]) There exist a constant η = η(X) which satisfies the following property. Let x ∈ X, elements g_{1}, ..., g_{ℓ} ∈ G generate a discrete group Γ and d(x, g_{j}(x)) ≤ η. Then the group Γ is a finite extension of a nilpotent subgroup Γ′ ⊂ Γ; the group Γ′ is generated by Γ ∩ U_{Z} ⊂ Isom(X), where U_{Z} is a Zassenhaus neighborhood in G.

Proof: We shall identify X with a Borel subgroup P of G. We can assume that U = U_{Z} is the ϵ-neighborhood U_{ϵ}(1) with respect to the metric on G. Since X is homogeneous we will identify the point x with the projection of 1 ∈ G to X.

Lemma 6.41 There exist numbers η < ϵ_{1} < ϵ and an integer N such that if g_{1}, ..., g_{ℓ} ∈ G are such that d(x, g_{j}(x)) ≤ η then:

(1) K has an ϵ_{1}/2–net of N elements;

(2) each word w = w(g_{1}, ..., g_{ℓ}) in the generators g_{1}, …, g_{ℓ} of the length ≤ N has the property: d(x, wx) ≤ ϵ_{1};

(3) for each w as above we have:

w(U_{3ϵ_{1}}(1))w^{−1} ⊂ U_{ϵ}(1)

Proof: We start with ϵ_{1} = ϵ/5. Find a number N such that (1) is satisfied, then choose η such that (2) holds. Now each word w above is the product pk where k ∈ K and d(p, 1) ≤ ϵ_{1} (accoridng to (2)). Therefore, if δ ∈ U_{3ϵ_{1}}(1) and τ := kδk^{−1} then τ ∈ U_{3ϵ_{1}} since the metric on G is bi-invariant under K. Thus

d(1, pkδk^{−1}p^{−1}) = d(1, pτp^{−1}) ≤ 2d(1, p) + d(1, τ) ≤ 5ϵ_{1} □

Let Γ′ denote the subgroup of Γ generated by all elements γ ∈ Γ ∩ U_{ϵ}(1) (apriori this subgroup can be trivial). Let

Γ = ⋃_{j=1}^{ν} γ_{j}Γ′

be the coset decomposition of Γ. We shall use l(·) to denote the word length on the group Γ (with respect to the generating set {g_{1}, ..., g_{ℓ}}).

Lemma 6.42 ν < ∞.

Proof: Let γ_{j} = w = g_{i_{1}}....g_{i_{M}} be a word of the length M > N. Then w = w_{1}w_{2} = w_{3}w_{4} where l(w_{2}) < l(w_{4}) ≤ N, w_{j} = p_{j}k_{j} and d(k_{2}, k_{4}) ≤ ϵ_{1}. Therefore,

δ = w_{4}w_{2}^{−1} = p_{4}k_{4}k_{2}^{−1}p_{2}^{−1}

However, d(1, k_{4}k_{2}^{−1}) ≤ ϵ_{1}. On another hand,

ϵ_{1} ≥ d(w_{j}x, x) = d(p_{j}x, x)

Therefore, d(1, δ) ≤ 3ϵ_{1}. This implies that

w = w_{3}δw_{2} , l(w_{3}w_{2}) < M

Consider the element

(w_{3}w_{2})^{−1}w = w_{2}^{−1}w_{3}^{−1}w_{3}δw_{2} = w_{2}^{−1}δw_{2}

Then the property (3) in Lemma 6.41 implies that this element belongs to U_{ϵ}(1) ∩ Γ ⊂ Γ′. Thus, w_{3}w_{2} and w belong to the same coset (mod Γ′), but the length of w_{3}w_{2} is strictly smaller than M. The induction argument thus implies that for all cosets (mod Γ′) we can find representatives γ_{j} such that l(γ_{j}) ≤ N. This implies that ν is finite. □

Thus |Γ : Γ′| < ∞. According to Theorem 6.39, the group Γ′ is nilpotent, therefore, Γ is finite extension of a nilpotent group. □

Other versions of this proof see in [197], or [16]. For generalization of this Theorem to the case of nonpositive sectional curvature see [84], [64].

Corollary 6.43 (D. Kazhdan and G. Margulis, [115]) There exist constants μ_{n} which satisfy the following property. Let x ∈ ℍ^{n}, elements g_{1}, ..., g_{k} ∈ Isom(ℍ^{n}) generate a discrete group Γ and d(x, g_{j}(x)) ≤ μ_{n}. Then the group Γ is almost Abelian and is a finite extension of an almost Abelian subgroup Γ′ ⊂ Γ; the group Γ′ is generated by Γ ∩ U_{Z} ⊂ Isom(ℍ^{n}), where U_{Z} is Zassenhaus neighborhood in Isom(ℍ^{n}).

Definition 6.44 The number μ := μ_{3} is called the Margulis constant for ℍ^{3}.

6.12 Geometric consequences of Kazhdan-Margulis-Zassenhaus lemma

Let Γ be a discrete subgroup of P SL(2, ℂ). Let Γ_{μ} denote the subset of ℍ^{3} consisting of all points z such that there exists an element of infinite order h ∈ Γ such that d(z, hz) ≤ μ. This set is Γ-invariant. The stabilizer of each component of Γ_{μ} is an elementary subgroup of Γ by Corollary 6.43. We now analyse these components assuming for simplicity that Γ is torsion-free (otherwise we first pass to a torsion-free finite index subgroup in the stabilizer of a component of Γ_{μ}). There are two cases to consider depending on whether the stabilizer of the component U ⊂ Γ_{μ} contains a loxodromic element. Since Γ is discrete, either the stabilizer of U consists only of loxodromic elements (Case 1) or only of parabolic elements (Case 2).

Case 1. U is a component of Γ_{μ} whose stabilizer Γ_{U} contains a loxodromic element. We will assume that U is not a single geodesic. The stabilizer Γ_{U} is a cyclic group generated by g which contains a loxodromic element h so that

d(x, h(x)) ≤ μ, for some x ∈ U

Note that it might happen that ℓ(g) > μ. Let r denote the distance between x and the axis A_{h} of h. The centralizer of h in Isom(ℍ^{3}) acts transitively Nbd_{r}(A_{h}). Thus for each y ∈ ∂Nbd_{r}(A_{h}) we have: d(y, h(y)) = d(x, h(x)) ≤ μ.

Exercise 6.45 If λ is the shortest geodesic arc between x and A_{h}, show that for each z ∈ λ we have: d(z, h(z)) ≤ d(x, h(x)). Hint: use convexity of the distance function between geodesics λ, h(λ) in ℍ^{3}.

Therefore

{y ∈ ℍ^{3} : d(y, h(y)) ≤ μ}

equals Nbd_{r}(A_{h}) for some r. Since

U = ∪_{h∈Γ_{U}} {y ∈ ℍ^{3} : d(y, h(y)) ≤ μ

it follows that U = Nbd_{r}(A_{g}) for some r. The domain U is precisely invariant under Γ_{U} in Γ and the projection of U to M(Γ) is the solid torus U/Γ_{U}. The quotient U/Γ_{U} ⊂ M(Γ) is called a Margulis tube in M(Γ).

Case 2. U is a component of Γ_{μ} whose stabilizer Γ_{U} is a parabolic subgroup. The same arguments as in the Case 1 show that for each h ∈ Γ_{U}

{y ∈ ℍ^{3} : d(y, h(y)) ≤ μ}

equals a horoball in ℍ^{3} centered at a point p ∈ ∂_{∞}ℍ^{3}. Therefore U itself is a horoball centered at p. There are two cubcases which can occur:

(2a) Suppose that Γ_{U} is an abelian group A of rank 2. The quotient of U by A is homeomorphic to T^{2} × ℝ_{+}, where T^{2} is the 2-dimensional torus.

(2b) Suppose that Γ_{U} is infinite cyclic group. The quotient of U by A is homeomorphic to D^{∗} × ℝ_{+}, where D^{∗} = {z ∈ ℂ : 0 < |z| ≤ 1} is the punctured disk. The product D^{∗} × ℝ_{+} is called a punctured solid cylinder.

In the cases (2a), (2b) we shall refer to the horoball U as a precisely invariant horoball, the projection of U to M(Γ) is called a Margulis cusp of rank r = 1, 2 (depending on the rank of Γ_{U}).

Exercise 6.46 Show that every parabolic fixed point p of Γ corresponds to a horoball B ⊂ ℍ^{3} with center at p which is precisely invariant under the stabilizer of p in Γ. The horoball B is a component of Γ_{μ}. This implies that every parabolic fixed point is never a conical limit point.

Thus Γ_{μ} is a disjoint union of horoballs and tubular neighborhoods of geodesics in ℍ^{3}. Stabilizers of each of these domains are elementary subgroups of Γ. The projection of Γ_{μ} to M(Γ) is denoted M_{(0,μ/2]}, it is called the “thin” part of the manifold M = M(Γ). The complement M_{(μ/2,∞)} := M − M_{(0,μ/2]} is called the “thick” part of M. The decomposition of M into the union of M_{(μ/2,∞)} and M_{(0,μ/2]} is called the thick-thin decomposition of M.

If Γ is torsion-free, then points of the thin part can be characterized by the property: the injectivity radius at each point of M_{(0,μ/2]} is at most μ/2.

Another description of cusped parabolic fixed points.

We again consider only torsion-free case, the general case is done by taking a torsion-free subgroup of finite index. Recall that if p is a fixed point of a rank 2 parabolic subgroup of Γ, then p is necessarily a cusped parabolic fixed point.

Exercise 6.47 Suppose that the stabilizer A of p in Γ has rank 1. Show that p is a cusped parabolic fixed point iff there exists a pair of disjoint disks D, D′ ⊂ \overline{ℂ} which are tangent at p so that D ∪ D′ is precisely invariant under the subgroup A of Γ. Hint: use the Exercise 6.46.

Let p ∈ \overline{ℂ} be a cusped parabolic fixed point as above. Without loss of generality we can assume that p = ∞, D, D′ are half-planes in ℝ^{2} given by the inequalities

D = {z ∈ ℝ^{2} : x_{2} > 1}, D′ = {z ∈ ℝ^{2} : x_{2} < −1}  (23)

Then we can consider a domain V (p) in ℝ_{+}^{3}

V (p) = {x ∈ ℝ_{+}^{3} : x_{2}^{2} + x_{3}^{2} > 1}  (24)

which is the convex hull of D ∪ D′ in ℍ^{3}. The union D ∪ D′ is precisely invariant iff V is precisely invariant. Domains like V (p) are called “cuspidal neighborhoods” of parabolic fixed points. By abusing notations we shall call the projection of V (p) to M̄(Γ) a cuspidal neighborhood corresponding to the Margulis cusp U/A.

Now we are interested in the intersection of precisely invariant horoballs for parabolic points (of rank 2) with the convex hulls C(Λ(Γ)). We again assume that p = ∞ is the fixed point for the group A generated by f : z ↦ z + 1 and h : z ↦ z + a. Suppose that the discrete group Γ is nonelementary, therefore there is at least one limit point z ∈ ℂ. Then C(Λ(Γ)) contains the convex hull C(Az) of the orbit Az. Let r denote the Euclidean radius of a maximal open disk contained in ℂ−Az. Suppose that x ∈ ℍ^{3} is such that d_{𝔼^{3}}(x, ℂ) > r, then x ∈ C(Az) (here d_{𝔼^{3}} is the Euclidean distance). Therefore C(Az) contains a precisely invariant horoball O with the center at ∞. Thus, the whole Margulis cusp O/A is contained in C(Γ) = C(Λ(Γ))/Γ.

Lemma 6.48 (1) There is 1-1 correspondence between the Margulis cusps of M(Γ) and Γ-conjugacy classes of maximal parabolic subgroups of Γ.

(2) Two Margulis cusps C_{1}, C_{2} in M(Γ) contain homotopically nontrivial homotopic loops iff C_{1} = C_{2}.

Proof: Suppose that γ_{j} ⊂ C_{j} are homotopic loops representing α_{j} ∈ π_{1}(C_{j}), j = 1, 2. Lift C_{j} to horoballs O_{j} in ℍ^{3} with the stabilizers A_{j} = π_{1}(C_{j}) ⊃ α_{j}. Then there is an element g ∈ Γ such that gα_{1}g^{−1} = α_{2}, therefore fixed point sets of A_{2} and gA_{1}g^{−1} coincide. Hence A_{2} and gA_{1}g^{−1} generate a parabolic subgroup in Γ. However the groups A_{2} and gA_{1}g^{−1} are maximal parabolic subgroups of Γ, thus A_{2} = gA_{1}g^{−1} and we conclude that C_{1} = C_{2}. □

6.13 Characterization of geometrically finite groups

Theorem 6.49 Let Γ ⊂ Isom(ℍ^{3}) be a discrete group. Then the following conditions are equivalent:

(i) Γ is geometrically finite (in the sense of the Definition 6.24).

(ii) The space CM(Γ)_{[μ,∞)} is compact.

(iii) Each limit point of Γ is either a point of approximation or a cusped parabolic point. (iv) Γ has a convex fundamental polyhedron with finitely many faces.

(v) Γ is finitely generated and V ol(CM(Γ)) < ∞.

Warning 6.50 If in the property (v) we do not assume Γ to be finitely generated, then it will fail to imply geometric finiteness. Moreover, there are examples (due to E. Hamilton [86]) of discrete subgroups Γ of Isom(ℍ^{4}) which cannot be finitely generated but V ol(Nbd_{ϵ}CM(Γ)) < ∞. These groups contain elliptic elements of arbitrarily high orders.

Proof: I give here only a partial proof of this theorem, for details see [14], [27] (the paper of Bowditch [29] also treats geometrical finiteness in the case of discrete groups acting on spaces of negatively pinched curvature). We will assume that Γ is torsion-free. We first prove (i) ⇒ (ii). Let δ < min(ϵ, μ) where μ is the Margulis constant. There exists a maximal set {x_{k} ∈ CM(Γ)_{[μ,∞)}, k = 1, ..., n} such that the balls B_{δ/2}(x_{k}) are disjoint (since the volume of Nbd_{ϵ}(CM(Γ)) is finite and each ball contributes a definite amount of volume). Thus by maximality the balls B_{δ}(x_{k}) cover the whole space C(Γ)_{[μ,∞)}.

Next we prove (ii) ⇒ (iii). Note that compactness of CM(Γ)_{[μ,∞)} implies that each parabolic fixed point of Γ is cusped. Suppose that we have a nonrecurrent geodesic ray ρ in CM(Γ). We can assume that ρ is entirely contained in a component of CM(Γ)_{(0,μ]}. This component cannot be compact. Therefore, a component of the preimage of ρ in ℍ^{3} is contained in a horoball with center at a parabolic fixed point. Hence each limit point is either a point of approximation or a cusped parabolic fixed point. This proves (iii).

Let’s prove the implication (iii) ⇒ (iv) assuming that Γ contains no parabolic elements. If Γ has a Dirichlet fundamental polyhedron D with infinitely many faces then any

point z of accumulation of faces cannot be a conical limit point, otherwise we can take a geodesic ray σ ⊂ D which is asymptotic to z. The Γ-orbit of this ray cannot intersect a compact set infinitely many times (by local finiteness of fundamental polyhedra). Recall that the collection of faces of D is locally finite in ℍ^{3}. Thus we have a seqeunce of faces S_{α} of D which is Hausdorff-convergent to z (in the metric which determines the topology of ℍ^{3} ∪ ∂_{∞}ℍ^{3}). The faces S_{α} pair with faces S′_{α} via pairing transformations g_{α} ∈ Γ and (after taking a subsequence) the faces S′_{α} Hausdorff-converge to a point w ∈ \overline{ℂ}. Then g_{α} converges to the quasiconstant w_{z} which means that z is a limit point of Γ. Contradiction. If there are parabolic points, then one has to work a bit more (see [14]). This proves (iii) ⇒ (iv).

Now suppose that a convex fundamental polyhedron D of Γ has only finitely many sides. Then any point of ∂_{∞}D is either parabolic fixed point or a point of discontinuity. Again let us assume that there are no parabolic fixed points. Thus ∂_{∞}D consists only of the points of discontinuity. This implies that to get the intersection D ∩ C(Λ(Γ)) we must “chop off” ∂_{∞}D via a hypersurface which has compact intersection with D. Thus D∩C(Λ) is compact, whence CM(Γ) (which is the quotient of D ∩ C(Λ(Γ))) is compact.

We refer to [28] for the proof of the implication (v) ⇒ (i). □

Definition 6.51 A discrete subgroup Γ ⊂ Isom(ℍ^{3}) is called convex- cocompact if it is geometrically finite and has no parabolic elements.

Corollary 6.52 A subgroup Γ ⊂ Isom(ℍ^{3}) is convex–cocompact if and only if CM(Γ) is compact.

Exercise 6.53 Prove that every finitely-generated discrete subgroup of Isom(ℍ^{2}) is geometrically finite.

Let (M, P) be a compact paired 3-dimensional manifold. Suppose that G ⊂ Isom(ℍ^{3}) is a (torsion-free) geometrically finite Kleinian group so that M ≅ Nbd_{ϵ}CM(G)_{[ν,∞)} for some ϵ > 0, 0 < ν < μ_{3} and this homeomorphism carries P onto Nbd_{ϵ}CM(G)_{[ν,∞)} ∩ Nbd_{ϵ}CM(G)_{(0,ν]}. Then we say that (M, P) admits a geometrically finite hyperbolic structure.

6.14 Convex hull and the domain of discontinuity

The boundary of the convex hull C(Λ(Γ)) has the property: each point x ∈ C(Λ(Γ)) is either contained in a complete hyperbolic geodesic γ ⊂ C(Λ(Γ)) or it lies in the interior of a convex hyperbolic polygon with ideal vertices. Connected components of the boundary of the convex hull C(Λ(Γ)) have the natural path-metric d_{C} induced from the ambient hyperbolic space. It turns out that the universal cover of each component of (∂C(Λ(Γ)), d_{c}) is isometric to the hyperbolic plane. This is quite obvious if C(Λ(Γ)) is bounded by a locally finite family of faces, the general case is more saddle and requires some approximation arguments (see [56]). In this section we consider the “nearest-point projection” from (ℍ^{3} ∪Ω(Γ))−C(Λ(Γ)) to ∂C(Λ(Γ)), our discussion mainly follows [55].

Suppose that C ⊂ ℍ^{3} is a closed convex nonempty subset (we will use C = C(Λ) or C = Nbd_{ϵ}C(Λ) ). Let Λ := ∂_{∞}C, Ω := \overline{ℂ} \ Λ.

Let r = r_{C} : ℍ^{3}−C → ∂C be the nearest-point projection, we know that it is a distancedecreasing mapping. Our goal is to extend r to the domain Ω = \overline{ℂ}−Λ. Choose a base-point x_{0} ∈ C, we will normalize all Busemann functions on ℍ^{3} at this point. According to the definition, for each x ∈ ℍ^{3} − C we have:

d(x, r(x)) = min{d(x, y)|y ∈ C}

Hence the point r(x) also gives minimum to the normalized distance function:

α_{x}(y) := d(x, y) − d(x_{0}, x); d(x, r(x)) = min{α_{x}(y)|y ∈ C}

The normalized distance functions α_{x}(y) have limits as x → ξ ∈ ∂_{∞}ℍ^{3}, the limits are the Busemann functions β_{ξ} normalized at x_{0}. Hence for each ξ ∈ Ω ⊂ ∂_{∞}ℍ^{3} we let

r(ξ) ∈ ∂C be the point such that β_{x}(r(x)) = min{β_{ξ}(y)|y ∈ C}

Then r = r_{C} : Ω ∪ ℍ^{3} \ C → ∂C is a continuous mapping. Geometrically one can describe the mapping r as follows. Let x ∈ ℍ^{3} \ C, take B(x) ⊂ ℍ^{3} be the largest metric ball with center at x such that its interior is disjoint from C. Since the distance function in ℍ^{3} is strictly convex, the ball B(x) touches ∂C at a single point and this point is r_{C}(x). The Chabauty-limits of metric balls in ℍ^{3} are horoballs, thus for each ξ ∈ Ω we take the maximal horoball B(ξ) with center at ξ whose interior is disjoint from C. Then r(ξ) is the point of where ∂C touches B(ξ).

Proposition 6.54 The map r : Ω → ∂C is onto.

Proof: Suppose that w ∈ ∂C. There is a supporting hyperplane P for C which contains the point w. This plane bounds two closed hyperbolic half-spaces P^{+} ⊃ C and P^{−}. Then int(P^{−}) does not intersect C and it is clear that there is a horoball B ⊂ P^{−} which touches the convex set C ∩P at the point w. We let ξ be the center of the horoball B, then r(ξ) = w. □

Consider the function d(x) = d(x, C).

Proposition 6.55 ([55]) The function d(x) is differentiable on ℍ^{3} − C.

Proof: Let y = r(x), x ∈ ℍ^{3} − C. Let P denote a supporting plane at y. Introduce two functions: f_{1}(z) = d(z, P), f_{2}(z) = d(z, y). Then f_{1} ≤ d ≤ f_{2}, f_{1}(x) = d(x) = f_{2}(x) and the derivatives of f_{1}, f_{2} at x coincide. Thus, d is a differentiable function at x. □

Corollary 6.56 Let ϵ > 0 and U_{ϵ} = Nbd_{ϵ}C be the ϵ–neighborhood of C. Then the boundary of U_{ϵ} is differentiable.

Lemma 6.57 If U is a closed convex smooth submanifold with boundary in ℍ^{3}, then r_{U} : Ω → ∂U is injective.

Proof: Suppose that r_{U}|_{Ω} is not injective. Then there are two distinct point ξ, η ∈ Ω such that the horoballs B(ξ), B(η) are tangent to ∂U at the same point y. These horoballs are distinct and their boundaries are not tangent at y (since U is a smooth submanifold). The sets U, B(ξ), B(η) are convex; thus there are (supporting) hyperplanes P_{ξ}, P_{η} ⊂ ℍ^{3} which “separate U from B(ξ), B(η) at x”, i.e.:

• U ⊂ P_{ξ}^{+}, U ⊂ P_{η}^{+};

• B(ξ) ⊂ P_{ξ}^{−}, B(η) ⊂ P_{η}^{−}

Here P_{η}^{±}, P_{ξ}^{±} are closed half-spaces bounded by P_{ξ}, P_{η}. Thus the smooth submanifold U has two distinct tangent planes P_{ξ}, P_{η} at the point x which is impossible. □

Corollary 6.58 Suppose that C = C(Λ) ⊂ ℍ^{3} is the convex hull of a closed set Λ ⊂ \overline{ℂ}, U = Nbd_{ϵ}C is the ϵ-neighborhood of C for some ϵ > 0. Then the function r_{U} : Ω → ∂U is a homeomorphism.

Proof: The function r_{U}|_{Ω} is injective, surjective and continuous. Both ∂U and Ω are 2dimensional manifolds. Thus r is a homeomorphism. □

As above let U be an ϵ-neighborhood of C, then we have three projections:

r_{U} : Ω → ∂U, π := r_{C}|_{∂U} : ∂U → ∂C, r_{C} : Ω → ∂C

Proposition 6.59 r_{C} = π ∘ r_{U}.

Proof: It is enough to prove that for each x ∈ ℍ^{3} − U we have π ∘ r_{U}(x) = r_{C}(x). Note that D := d(x, r_{U}(x)) + d(r_{U}(x), π(r_{U}(x))) ≥ d(x, r_{C}(x)). Suppose that there is a point y ∈ ∂C such that d(x, y) < D. Then the geodesic segment [xy] intersects ∂U in a point z such that:

d(x, z) ≥ d(x, ∂U), d(z, ∂C) ≥ ϵ = d(r_{U}(x), π(r_{U}(x)))

Contradiction. □

From now on we restrict ourselves to the case when C has nonempty interior. Whence C is a (topological) submanifold with boundary in ℍ^{3}. Clearly the projection r_{C} : cl(U −C) → ∂C is a deformation retraction and the inclusion U − C → cl(U − C) is a homotopyequivalence. On the other hand, U − C ≅ (0, ϵ] × ∂U. We conclude that r_{C} : Ω → ∂C is a homotopy-equivalence. Suppose that Γ is a discrete subgroup of Isom(ℍ^{3}) which keeps Λ = ∂_{∞}C invariant, hence Λ(Γ) ⊂ Λ. Then Ω = \overline{ℂ} \ Λ ⊂ Ω is Γ-invariant as well, we shall assume that Γ acts freely on Ω. The retractions r_{C}, r_{U} are Γ-invariant, thus we have a retraction

r̄: Ω/Γ → ∂C/Γ

By the same arguments as above we conclude that r̄ is a homotopy-equivalence. However most of the time the mapping r̄ is very far from being a homeomorphism since C has “corners”. Nevertheless we will find a portion of Ω where r_{C} is injective. Consider the upper half-space model ℝ_{+}^{3} = ℂ × ℝ_{+} for ℍ^{3}, where ℝ_{+} := {t : t > 0}; we assume that Λ ⊂ \overline{ℂ} contains the point ∞ and Λ ∩ ℂ lies in \{z : Im(z) ≤ c\}. Suppose also that the region

L = L_{c,a} := {(z, t) ∈ ℝ_{+}^{3} : Im(z) = c, t ≥ a > 0}

lies entirely in ∂C. (Note that L is the intersection of a horoball centered at ∞ and geodesic hyperplane asymptotic to ∞.)

Let D_{+} := {z : Im(z) ≥ c + a}. It is easy to see that the restriction of r_{C} to D_{+} is the Euclidean rotation in 𝔼^{3} by π/2 around the line {(z, t) : t = 0, Im(z) = c}. Hence r_{C} : D_{+} → ∂C is injective, moreover it is a conformal mapping between D_{+} and r(D_{+}) ⊂ ∂C, where the conformal structure on ∂C is given by the path-metric induced from ℍ^{3}.

Now we assume that Ω is a component of Ω(Γ), Γ_{Ω} is the stabilizer of Ω in Γ and S := Ω/Γ_{Ω} is a Riemann surface of finite type (i.e. it is obtained from a compact Riemann surface by removing a finite number of points: punctures of S). According to Theorem 6.37, elements of π_{1}(S) corresponding to loops going around punctures, are mapped to parabolic elements of Γ under the epimorphism π_{1}(S) → Γ. Let γ be one of the parabolic elements corresponding to a puncture P. We assume that ∞ ∈ \overline{ℂ} is the fixed point of γ; thus after lifting a punctured disc around P into Ω (and conjugating Γ by a Euclidean rotation) we get a domain D ⊂ ℂ which is precisely invariant under A := ⟨γ⟩ in Γ and D contains the region D_{q} := {z : Im(z) ≥ q} for some q. The element γ acts on ℂ as the translation z ↦ z + a. D_{q} ∩ Λ = ∅, let D_{c} := {z : Im(z) ≥ c} denote the maximal closed Euclidean half-plane in ℂ whose interior is disjoint from Λ. Its boundary {z : Im(z) = c} contains at least one point x ∈ Λ, therefore it also contains the whole A-orbit of x. This means that C(Λ) ∩ {(z, t) : Im(z) = c, t ∈ ℝ_{+}} contains the domain L := {(z, t) : Im(z) = c, t ≥ a}. This is the situation that we had analysed above. Our conclusion is the following:

Proposition 6.60 Assume that S = Ω/Γ_{Ω} is a Riemann surface of finite type and Γ_{Ω} acts freely on Ω. Then there exists a collection of disjoint punctured discs D_{j} around punctures of S so that restriction of the retraction r̄: S → ∂C(Λ(Γ))/Γ to the union ∪_{j}D_{j} is injective and conformal.

Hence r̄ : S → ∂C(Λ(Γ))/Γ_{Ω} is a homotopy-equivalence onto its image which is a homeomorphism near punctures of S. Nielsen Realization Theorem implies that the surfaces S, r̄(S) are quasiconformally homeomorphic via a homeomorphism homotopic to r̄. Moreover, r̄(Ω) is a component of ∂C(Λ(Γ_{Ω})).

Corollary 6.61 If Ω/Γ_{Ω} has finite conformal type, then Σ = ∂C(Λ(Γ_{Ω}))/Γ_{Ω} has finite hyperbolic area 2π|χ(Σ)|.

A Kleinian group Γ ⊂ P SL(2, ℂ) is said to be analytically finite if it is nonelementary and Ω(Γ)/Γ is a finite collection of Riemann surfaces of finite conformal type.

Corollary 6.62 Suppose that Γ ⊂ P SL(2, ℂ) is analytically finite and Λ(Γ) is not contained in a round circle in 𝕊_{∞}^{2}. Then the manifolds C(Λ(Γ))/Γ and M̄(Γ) are homeomorphic. The homeomorphism carries Margulis cusps to Margulis cusps.

Proof: We already know that the complement M̄(Γ) − intC(Λ(Γ))/Γ is naturally homeomorphic to I × Ω(Γ)/Γ. This homeomorphism carries the intersection of Margulis cusps of M(Γ) with M̄(Γ) − intC(Λ(Γ))/Γ to P × I, where P is a union of punctured discs around punctures of Ω(Γ)/Γ. Since Λ(Γ) is not contained in a round circle, the convex hull C(Λ(Γ)) and the convex core CM(Γ) are a 3-dimensional manifolds (with boundary). Attaching M̄(Γ) − intC(Λ(Γ))/Γ to CM(Γ) amounts to attaching a collar to CM(Γ) which does not change the topology. □

Corollary 6.63 A discrete subgroup Γ ⊂ Isom(ℍ^{3}) is convex–cocompact if and only if \overline{M}(Γ) is compact.

More generally,

Corollary 6.64 (A. Marden, [129]) A discrete group G ⊂ Isom(ℍ^{3}) is geometrically finite if and only if there is a finite disjoint collection U of cuspidal neighborhoods of cusps in M̄(Γ) so that the complement

M̄(Γ) − int(U)

is compact.

6.15 Combination Theorems

Combination theorems is a convenient tool to construct a great deal of examples of Kleinian groups. In this section we describe several such theorems, the first of them was discovered by Klein in the late 19-th century, the other combination theormes were proven by Maskit in 1960-s.

6.15.1 Klein Combination

We start with a simple example of a combination theorem which is called the “Klein Combination.” Suppose that G_{1}, G_{2} are two Kleinian groups in Isom(ℍ^{3}) with the fundamental domains D_{1}, D_{2} ⊂ \overline{ℂ}, which satisfy the following condition:

intD_{1} ⊃ \overline{ℂ} − D_{2}, intD_{2} ⊃ \overline{ℂ} − D_{1}  (25)

See the Figure 8.

Figure 8: Klein Combination.

Theorem 6.65 Under the conditions above, the group G = ⟨G_{1}, G_{2}⟩ ⊂ Isom(ℍ^{3}) generated by G_{1}, G_{2} is a Kleinian group which is isomorphic to the free product G_{1}∗G_{2}. A fundamental domain for the action of G on the complex plane is D = D_{1} ∩ D_{2}.

Proof: Choose any point x ∈ D and apply to x any reduced word

w = g_{1}g_{2}...g_{n−1}g_{n}

where g_{even} ∈ G_{2}, g_{odd} ∈ G_{1}. Then the induction argument shows that w(x) ∉ D. For details see [133]. □

A typical example is the Schottky group where we iterate application of Theorem 6.65.

Proposition 6.66 Suppose that G ⊂ Isom(ℍ^{3}) is a nonelementary group (not necessarily discrete). Then G contains a nonabelian Schottky subgroup.

Proof: Since G is nonelementary it must contain two loxodromic elements g, h whose fixedpoint sets F ix(g), F ix(h) ⊂ \overline{ℂ} are disjoint. Let D, E be two sufficiently small disjoint closed round disks centered at the repulsive fixed points x, y of g, h. The convergence property of Moebius transformations implies that we can find n ≫ 1 so that g^{n}(D) ⊃ D ∪ E, h^{n}(E) ⊃ D ∪ E and the exteriors D′ := \overline{ℂ} − g^{n}(D), E′ := \overline{ℂ} − h^{n}(E) are disjoint. Thus the complement to D ∪ E ∪ D′ ∪ E′ is a fundamental domain for the Schottky group ⟨g^{n}, h^{n}⟩. □

Similarly we get:

Proposition 6.67 Suppose that G is a discrete subgroup of Isom(ℍ^{3}), h ∈ Isom(ℍ^{3}) is a loxodromic elements whose fixed point set F ix(h) is disjoint from the limit set Λ(G). Then:

1. If for every element g ∈ G we have: g(F ix(h)) ∩ F ix(h) = ∅, then there is n ≫ 1 so that the subgroup of Isom(ℍ^{3}) generated by G, ⟨h^{n}⟩ ⊂ Isom(ℍ^{3}) is a Kleinian subgroup isomorphic to the free product G ∗ ⟨h^{n}⟩.

2. There is a finite-index subgroup G^{0} ⊂ G and n ≫ 1 so that the subgroup of Isom(ℍ^{3}) generated by G^{0}, ⟨h^{n}⟩ ⊂ Isom(ℍ^{3}) is a Kleinian group isomorphic to the free product G^{0} ∗ ⟨h^{n}⟩.

Proof: Consider (1). Since g(F ix(h)) ∩ F ix(h) = ∅ for every g ∈ G, we can find a fundamental domain D_{1} for the action of G in \overline{ℂ} so that F ix(h) ⊂ D_{1}. As in the proof of Proposition 6.66, if n is sufficiently large, we find a fundamental domain D_{2} := \overline{ℂ}−(D ∪D′) for the cyclic group ⟨h^{n}⟩ so that: \overline{ℂ} − D_{1} ⊂ D_{2}, \overline{ℂ} − D_{2} ⊂ D_{1}. Hence we can apply Klein Combination Theorem, which implies the Assertion (1).

Now consider (2). Since F ix(h) ∩ Λ(G) = ∅, there is only a finite number of elements g_{1}, ..., g_{m} ∈ G − {1} such that g_{j}(F ix(h)) ∩ F ix(h) ≠ ∅. By Selberg Lemma (Theorem 6.5) there exists a finite-index subgroup G^{0} ⊂ G which does not contain any of the elements g_{1}, ..., g_{m}. Hence we can apply Assertion (1). □

6.15.2 Maskit Combination

Before starting discussion of this type of combination theorems I would like to give a topological motivation for conditions which will appear below. Let M be a P L-manifold which is the union of two submanifolds with boundary A, B such that A ∩ B = S is a connected hypersurface and embedding of each A, B, S into M induces a monomorphism of the fundamental groups. According to Seifert– Van Kampen theorem, the fundamental group of M splits as π_{1}(A) ∗_{π_{1}(S)} π_{1}(B). Let Ã, B̃, S̃ denote the universal covers of A, B, C. Now we lift the decomposition A ∪_{S} B of M to the universal cover M̃. We get a tessellation of M̃ by copies of the universal covers Ã, B̃ separated by copies of S̃. Choose two representatives X, Y of Ã, B̃ separated by a copy Z of S̃. The stabilizers of X, Y in G = π_{1}(M) are the groups G_{A}, G_{B} which are isomorphic to π_{1}(A), π_{1}(B). The surface Z is precisely invariant under J = π_{1}(S) in both G_{A} and G_{B} (see Definition 6.8). Let X^{+} (resp. Y^{+}) be the component of M̃ − Z which contains X (resp. Y). It follows that the domain X^{+} is precisely invariant under J in G_{B} and Y^{+} is precisely invariant under J in G_{A}. The idea behind Maskit Combination Theorems is to reverse this picture. Namely, one starts with a pair of groups G_{A}, G_{B} acting on a space W so that G_{A} ∩ G_{B} contains a subgroup J. Then one finds a decomposition of W into domains similar to X^{+}, Y^{+} which are separated by Z so that Z is stabilized by J and X^{+}, Y^{+} have precise invariance properties as above. From this one would like to conclude that the group G = ⟨G_{A}, G_{B}⟩ generated by G_{A} and G_{B} is discrete, isomorphic to G_{A} ∗_{J} G_{B} and has a fundamental domain of special kind (or/and the quotient space of the group G splits like in Seifert– Van Kampen theorem). This is briefly the 1-st combination theorem. The second combination theorem treats the case when G is an HNN-extension. On the level of Seifert– Van Kampen theorem this corresponds to a nonseparating hypersurface S. I recommend you to consider this case yourself, find a decomposition of the space M̃ and conclude which domains in this decomposition have precise-invariance properties under subgroups of G. Then, read our formulation of the 2-nd Maskit Combination Theorem or Maskit’s book [133].

There are many different versions of Maskit Combination Theorems depending on how much properties we require from Kleinian groups and how strong conclusion we want. We shall present here four particular cases which will suffice for our needs.

Theorem 6.68 (First Maskit Combination Theorem: quasifuchsian amalgamation.) Let G_{2}, G_{1} be a pair of Kleinian groups such that G_{2} ∩ G_{1} ⊃ H. Suppose that H is a quasi-Fuchsian group with the limit set Λ(H) so that \overline{ℂ} − Λ(H) = Ω_{2} ∪ Ω_{1}. Assume that the domain Ω_{j} is precisely invariant under H in G_{j} (j = 1, 2). Then the group G generated by G_{1}, G_{2} is isomorphic to G_{1} ∗_{H} G_{2} and is discrete. If G_{1}, G_{2} are geometrically finite, then so is G. Under the isomorphism G → G_{1} ∗_{H} G_{2}, the image of any parabolic element of G is either conjugate into one of the groups G_{1}, G_{2} or it commutes with a parabolic element of a conjugate of H. The surface S(G) = Ω(G)/G is naturally conformally equivalent to (S(G_{1}) − Ω_{1}/H) ∪ (S(G_{2}) − Ω_{2}/H).

Proof: The proof can be found in [133, Chapter VII]. One way to prove this theorem is as follows. Let me assume that H is not contained as a subgroup of the index 2 in any subgroup of G_{1}, G_{2} and that H does not contain parabolic elements.

Precise invariance of Ω_{j} implies that the convex hulls C(Λ(G_{j})) intersect along a simplyconnected region Σ in ℍ^{3} which is the convex hull of Λ(H) so that Σ/H ≅ Ω_{1}/H × [0, 1]. In general it is not true that Σ is precisely invariant in both G_{1}, G_{2} under H. However, there exists a unique least area surface S′ ⊂ Σ/H (see [203]) and the projections of S′ to both C(Λ(G_{j}))/G_{j} are embeddings (see [62]). Then we take a lift S̃′ of S′ in Σ. Another way to argue is to use harmonic functions as in [150]:

Lemma 6.69 Consider the characteristic function λ : \overline{ℂ} → \{0, 1\} of the domain Ω_{1}. Let h = ext(λ) be the harmonic extension of λ to ℍ^{3} as in Section 4.11. Then the level set S̃ := {x|h(x) = 0.5} is precisely invariant under H in the both groups G_{1}, G_{2}.

Proof: If x ∈ ℍ^{3}, S_{x} is the unit tangent sphere at x, we define the “exponential map”

exp_{x} : S_{x} → 𝕊_{∞}^{2}

by sending each unit tangent vector v ∈ S_{x} to the geodesic ray ρ emanating from x in the direction v; the ray ρ determines a point at infinity ξ = exp_{x}(v). Since λ is H-invariant, S̃ is H-invariant as well (see Property 3 in Section 4.11). Suppose that there is an element g ∈ G_{1} − H such that x ∈ S̃ ∩ gS̃. By construction of ext(λ), the value h(x) is the Lebesgue measure of the domain exp_{x}^{−1}(Ω_{1}) on the unit tangent sphere S_{x}. Thus on the sphere S_{x} we get two domains: exp_{x}^{−1}(Ω_{1}), exp_{x}^{−1}(gΩ_{1}), both having 1/2 of the total area of S_{x}. These domains must be disjoint since Ω_{1} is precisely invariant under G_{1}. Hence Ω_{2} = gΩ_{1} and the group ⟨H, g⟩ is a ℤ/2-extension of H in G_{1}, which contradicts our assumptions. Similarly we prove that S̃ is precisely invariant under G_{2}. □

The level set S̃ could be singular, but by Sard’s theorem nearby there is a regular level set S̃′ := h^{−1}(0.5 ± ϵ) which is still precisely invariant. Note that the maximum principle for harmonic functions implies that S̃′ has no compact components. It seems to be true that the surface S̃′ has to be connected, but I do not know how to prove this. It is clear however that any connected component of S̃′ is H-invariant, so we retain the notation S̃′ for such connected component.

Then for each j the G_{j}–orbit of S̃′ is a disjoint family of connected surfaces that cuts from C(Λ(G_{j})) an open domain D_{j} that is invariant under G_{j}. We construct a graph T whose vertices are the domains gD_{j}, g ∈ G, j = 1, 2. Two vertices are connected by an edge iff either the corresponding domains are adjacent along a translate of S̃′. Then the precise invariance of S̃′ in both G_{j} implies that:

(a) The graph T is a tree which is properly embedded in ℍ^{3}.

(b) Stabilizers of vertices are conjugates of the groups G_{j} and stabilizers of edges are the conjugates of H in G.

We conclude that the group G is discrete and has the structure of amalgamated free product G_{1} ∗_{H} G_{2} (see Section 11.2).

Moreover there is a natural isometry between the manifold M(G) = ℍ^{3}/G and the manifold obtained by gluing D_{1}/G_{1} to D_{2}/G_{2} along S = S̃′/H (the gluing map is determined by the property that it is covered by the identity map S̃′ → S̃′). This isometry extends to a conformal diffeomorphism between M̄(G) and

[D_{1}/G_{1} ∪ (S(G_{1}) − Ω_{1}/H)] ∪_{S} [D_{2}/G_{2} ∪ (S(G_{2}) − Ω_{2}/H)]

This implies the rest of the assertions of the Theorem. □

To illustrate the first combination theorem consider the following example. Let M_{1} be a compact convex hyperbolic 3-manifold with a single boundary component S which is totally-geodesic. Let M_{2} be another copy of M_{1}, now “double” M_{1} along S by gluing M_{1} to M_{2} via the identity map S → S. The result is a closed hyperbolic 3-manifold M. Let Ω(G_{1}) be the domain of discontinuity for the action of G_{1} = π_{1}(M_{1}) on ℍ^{3}. The components of Ω(G_{1}) are round disks. Let Ω_{1} be one of these disks, its stabilizer in G_{1} is a Fuchsian subgroup H isomorphic to π_{1}(S). Then Ω_{1} is precisely invariant under H in G_{1}. Let σ be the inversion in the boundary circle of Ω_{1}. Define G_{2} as σG_{1}σ, Ω_{2} := σ(Ω_{1}). It is clear now that the tripe (G_{1}, G_{2}, H) satisfies the assumption of Theorem 6.68.

Theorem 6.70 (Second Maskit Combination Theorem: quasifuchsian amalgamation.) Let G_{0} be a Kleinian group such that G_{0} ⊃ H_{1}, H_{2}, where H_{j} are quasi-Fuchsian groups which stabilize different connected components Ω_{1}, Ω_{2} of Ω(G_{0}). Let A ∈ Isom(ℍ^{3}) be an element such that A(Ω_{1}) = \overline{ℂ} − cl(Ω_{2}) and AH_{1}A^{−1} = H_{2} induces an isomorphism ϕ of H_{1} and H_{2}. Then the group G generated by G_{0} and A is discrete and isomorphic to the HNN extension G_{0}∗_{ϕ} of G_{0} via ϕ. If G_{0} is geometrically finite, then so is G. Under the isomorphism G → G_{0}∗_{ϕ}, the image of any parabolic element in G is either conjugate into the group G_{0} or it commutes with a parabolic element of a conjugate of H_{1}. The surface S(G) = Ω(G)/G is naturally conformally equivalent to S(G_{0}) − (Ω_{1}/H_{1} ∪ Ω_{2}/H_{2}).

The proof can be found in [133, Chapter VII] and it is essentially the same as in the case of the first combination (with slight modifications you can repeat the arguments from the above proof of Theorem 6.68).

Figure 9: The 1-st Maskit Combination with cyclic amalgamation.

Now we describe two combination theorems where instead of quasifuchsian amalgamated subgroups we take cyclic subgroups. These cases of combination are easier to visualize and we will use them in Section 19.7 in the case when amalgamated subgroups are elliptic. Recall that if G acts on a set X and S ⊂ X then G(S) denotes the G-orbit of S in X.

Theorem 6.71 (First Maskit Combination Theorem: cyclic amalgamation.) Let G_{1}, G_{2} be a pair of Kleinian groups such that G_{1} ∩ G_{2} = H, where H is a cyclic subgroup. Let Φ_{j} be fundamental domains for the actions of G_{j} on \overline{ℂ} (j = 1, 2). Let B_{1}, B_{2} be open disks in \overline{ℂ} so that: J = cl(B_{1}) ∩ cl(B_{2}) = ∂B_{1} = ∂B_{2} is a topological circle. Suppose that:

• B_{j} is precisely invariant under H in G_{j} (j = 1, 2);

• Φ′_{j} := Φ_{j} ∩ G_{j}(B_{j}) ⊂ B_{j} (j = 1, 2);

• Φ′_{1} ∩ Φ_{2}, Φ_{1} ∩ Φ′_{2} have nonempty interiors.

Then the subgroup G ⊂ Isom(ℍ^{3}) generated by G_{1}, G_{2} is Kleinian and is isomorphic to G_{1} ∗_{H} G_{2}. If G_{1}, G_{2} are geometrically finite, then G is also geometrically finite. The quotient Ω(G)/G is naturally conformally isomorphic to

Ω(G_{1} − G_{1}(B_{1}))/G_{1} ∪_{L} Ω(G_{2} − G_{2}(B_{2}))/G_{2}

where gluing is along L = [J ∩ Ω(H)]/H. Any parabolic element in G is conjugate either into G_{1} or into G_{2} or it is conjugate to an element commuting with a parabolic element of H.

Theorem 6.72 (Second Maskit Combination Theorem: cyclic amalgamation.) Let G_{0} be a Kleinian group, H_{1}, H_{2} be its cyclic subgroups. Let Φ_{0} be a fundamental domain for the actions of G_{0} on \overline{ℂ}. Let B_{1}, B_{2} be open disks in \overline{ℂ} and A ∈ Isom(ℍ^{3}) be a Moebius transformation so that AH_{1}A^{−1} = H_{2}, this conjugation induces an isomorphism ϕ : H_{1} → H_{2}.

Suppose that:

• B_{j} is precisely invariant under H_{j} in G_{0} (j = 1, 2);

• A(B_{1}) ∩ B_{2} = ∅, A(∂B_{1}) ∩ ∂B_{2} = J is a topological circle;

• gB_{1} ∩ B_{2} = ∅ for all g ∈ G_{0};

• Φ_{0} ∩ (\overline{ℂ} − G_{0}(B_{1} ∪ B_{2})) has nonempty interior.

Then the subgroup G ⊂ Isom(ℍ^{3}) generated by G_{0}, A is Kleinian and is isomorphic to the HNN extension G_{0}∗_{ϕ:H_{1}→H_{2}} of G_{0} via ϕ. If G_{0} is geometrically finite, then G is also geometrically finite. The quotient Ω(G)/G is naturally conformally isomorphic to

∼ \[Ω(G_{0}) − G_{0}(B_{1} ∪ B_{2})]/G_{0}

where the identification is such that [J ∩ Ω(H_{2})]/H_{2} is identified with [A^{−1}(J) ∩ Ω(H_{1})]/H_{1} via the projection of A. Any parabolic element in G is conjugate either into G_{0} or it is conjugate to an element commuting with a parabolic element in H_{j}.

Figure 10: The 2-nd Maskit Combination with cyclic amalgamation.

Now I give two examples of the combination with cyclic amalgamation.

Example 6.73 Take A to be the rotation of the order n > 1 around the origin in ℂ; R := {z : 0.5 < |z| < 2} is a ring. A fundamental domain for the group J = ⟨A⟩ is the cone P := {z : 0 < arg(z) < 2π/n}. Pick two Kleinian groups Γ_{1}, Γ_{2} ⊂ Isom(ℍ^{3}) with the fundamental domains F_{1}, F_{2} such that ℂ − F_{1} is contained in P ∩ {z : |z| < 0.5}, ℂ − F_{2} is contained in P ∩ {z : |z| > 2} ∪ {∞}. See Figure 9, where Γ_{1}, Γ_{2} are cyclic groups generated by loxodromic elements g_{j}. Now let G_{j} = ⟨Γ_{j}, J⟩ be the subgroups of Isom(ℍ^{3}) generated by G_{j} and J (j = 1, 2). The group G generated by G_{1}, G_{2} is the result of the 1-st Maskit Combination of G_{1}, G_{2}, in particular G ≅ G_{1} ∗_{J} G_{2}. The precisely invariant disks are {|z| > 1} ∪ {∞}, {|z| < 1}.

Example 6.74 Start with a Fuchsian group G_{0}, it keeps the exterior E of the unit disk invariant. A fundamental domain Φ_{0} of G_{0} is the exterior of a collection of discs covering the unit circle, this domain and side-pairing are described in the Figure 10. Let H_{1}, H_{2} be maximal parabolic subgroups of G_{0} which are not conjugate in G_{0}. We identify E with the hyperbolic plane, then each H_{j} has a closed precisely invariant horoball B_{j} ⊂ E, G_{0}-orbits of these horoballs are disjoint. There exists a Moebius transformation A : B_{1} → cl(\overline{ℂ} − B_{2}) which conjugates H_{1} to H_{2}. Let G := ⟨G_{0}, A⟩, this group is the result of the 2-nd Maskit Combination.

6.16 Ahlfors finiteness theorem and its companions

Recall that a Riemann surface X is said to be of finite type if it can be obtained from a compact complex curve by removing a finite number of points. The following theorem is one of the most fundamental facts about finitely–generated Kleinian subgroups of Isom(ℍ^{3}).

Theorem 6.75 (L. Ahlfors) Ahlfors finiteness theorem: Let G be a nonelementary finitely generated Kleinian subgroup of Isom(ℍ^{3}). Then S(G) = Ω(G)/G is a finite union of Riemann surfaces of finite hyperbolic type.

Proof see in [3], [120]. This theorem has several extensions that we shall discuss later.

Corollary 6.76 Suppose that G is a finitely generated nonelementary Kleinian subgroup of Isom(ℍ^{3}) which does not contain parabolic elements. Then the boundary ∂C(Λ(G)) of the convex core CM(G) is a compact surface.

Proof: If the surface S(G) = Ω(G)/G has a puncture, then this puncture corresponds to a parabolic element of G (see Corollary 6.38). Thus S(G) is a compact surface. Recall that either C(Λ(G)) has empty interior (in which case CM(G) is compact) or ∂CM(G) is homeomorphic to S(G). □

Theorem 6.77 (W. Thurston) Let G be a geometrically finite Kleinian subgroup of Isom(ℍ^{3}) (i.e. Ω(G) ≠ ∅). Suppose that F is any finitely generated subgroup of G. Then F is geometrically finite.

Proof: I give here a proof only in the case when G has no parabolic elements, for the general case see [150]. Then N_{G} := C(Λ(G)/G) is compact and diam(N_{G}) = d < ∞. Without loss of generality we can assume that int(C(Λ(G)) ≠ 0, otherwise G is either elementary or is conjugate to a subgroup of P SL(2, ℝ), in which case all finitely-generated groups are geometrically finite and there is nothing to prove. Since diam(N_{G}) = d, for each point x ∈ N_{G} there exists a path ℓ of the length at most d which connects x and ∂N_{G}. Now consider the covering p : ℍ^{3}/F → ℍ^{3}/G. For each point y ∈ p^{−1}(x) a component ν of the path p^{−1}(ℓ) connects y and p^{−1}∂N_{G}. But p^{−1}∂N_{G} ∩ intN_{F} = ∅, where C(Λ(F))/F = N_{F}. Therefore, ν must intersect ∂N_{F} which is compact. Hence dist_{H}(N_{F} , ∂N_{F}) ≤ d, diam(N_{F}) < ∞, it follows that N_{F} is compact. □

Corollary 6.78 Suppose that in the situation above F is the stabilizer in G of a component Ω_{0} of Ω(G). Then F is geometrically finite.

Proof: Let S_{0} = Ω_{0}/F. Then, according to Ahlfors’ finiteness theorem, π_{1}(S_{0}) is finitely generated. Thus the group F (which is the quotient π_{1}(S_{0})) of must be finitely generated as well. □

6.17 Extensions of the Ahlfors Finiteness Theorem

Recall that a cusp of the hyperbolic manifold M(G) = ℍ^{3}/G is a subset which is the quotient of a horoball in ℍ^{3} by a maximal parabolic subgroup of G. Algebraically cusps 1-1 correspond to G-conjugacy classes of maximal parabolic subgroups of G (see 6.48).

Theorem 6.79 Sullivan finiteness Theorem: Let G be a discrete finitely generated subgroup of Isom(ℍ^{3}), then G has only finitely many cusps.

Analytical proofs see in [188], [121], topological proof is in [60], [122]. Note that purely topological methods cannot prove finiteness of the number of simply-connected and annular components of S(G) (see [106]).

Theorem 6.80 Ahlfors recurrence theorem: Let G be a finitely generated discrete subgroup of Isom(ℍ^{3}). Then the action of G on Λ(G) is recurrent, i.e. for every Lebesgue measurable subset A ⊂ Λ(G) of nonzero planar measure, there are infinitely many elements g ∈ G such that mes(A ∩ gA) ≠ 0.

We shall prove this result later, in Section 9.11.

Theorem 6.81 (Scott compact core theorem, [177, 178]). Suppose that M is a 3-manifold with the finitely generated fundamental group. Then: (a) the group π_{1}(M) is finitely presentable. (b) There exists a compact submanifold M_{c} in M so that the inclusion M_{c} ↪ M, is a homotopy-equivalence.

The manifold M_{c} is called a Scott compact core of the manifold M. A stronger version of Scott compact core theorem, that we shall need later, was proven by D. McCullough [138] (see also [122]):

Theorem 6.82 Suppose that M is a 3-manifold with with boundary and π_{1}(M) is finitely generated; Q ⊂ ∂M is a compact incompressible subsurface. Then one can choose a Scott compact core M_{c} ⊂ M such that

M_{c} ∩ ∂M = Q

CHECK IT!!!!

Scott compact core is unique up to isotopy:

Theorem 6.83 (D. McCullough, A. Miller and G. Swarup [139].) Let M be a P^{2}-irreducible, noncompact 3-manifold with finitely generated fundamental group. Suppose that N_{1} and N_{2} are irreducible Scott compact cores of M. Then N_{1} is homeomorphic to N_{2}.

Theorem 6.84 (Feighn–Mess finiteness theorem, see [127]). Let G be a finitely generated discrete subgroup of Isom(ℍ^{3}). Then G has only finitely many conjugacy classes of finite order elements.

This theorem is proven in [127] in two different ways. First is via the “Smith’s theory”, the second proof is based on the following

Theorem 6.85 (M. Feighn, G. Mess) Suppose that N is an open 3-manifold with the finitely generated fundamental group, F is a finite group of PL homeomorphisms acting on N. Then there exists a Scott compact core N_{c} ⊂ N which is invariant under F.

To derive Theorem 6.84 note that G contains a normal torsion-free subgroup of finite index Γ. Let N := ℍ^{3}/Γ. Then the finite group F = G/Γ acts on N with the quotient M(Γ). Thus F acts on N_{c} with a compact quotient which implies that G has only finitely many conjugacy classes of finite order elements.

Remark 6.86 This is unknown if each finitely generated discrete subgroup of Isom(ℍ^{n}) acts recurrently on its limit set (for n ≥ 4). The Ahlfors Finiteness Theorem and the rest of its “companions” fail for discrete subgroups of Isom(ℍ^{4}) even in the topological setting. See [112], [111], [168], [30].

6.18 Limit sets of geometrically finite groups

Theorem 6.87 (L. Ahlfors). Let G ⊂ Isom(ℍ^{3}) be a geometrically finite group. Then either the limit set of G is \overline{ℂ} or it has Lebesgue measure zero. In any case there is no nonconstant G-invariant measurable functions on Λ(G).

Proof: Recall that each limit point of G is either a point of approximation or a cusped parabolic limit point. In particular, a.e. limit point of G is a point of approximation. Let h : \overline{ℂ} → ℝ be any measurable function invariant under G. There exists a harmonic extension ext(h) : ℍ^{3} → ℝ (see Section 4.11). It follows that ext(h) is G-invariant. For almost every limit point z ∈ Λ(G) we have: lim_{w→z}^{c} ext(h) = h(z). Here lim_{w→z}^{c} means the conical limit, i.e w is convergent to z inside of a neighborhood of some geodesic.

Since ext(h) is G–invariant, we conclude that ext(h)(0) = h(z) for each 0 ∈ ℍ^{3} and almost every conical limit point z ∈ Λ(Γ). By (a.e.) continuity of ext(h) we conclude that h(p) = h(z) for almost every p ∈ \overline{ℂ}. This proves that h must be a.e. constant.

So, suppose that mes(Λ(G)) ≠ 0 and Λ(G) ≠ \overline{ℂ}. Then we define a measurable Ginvariant function h on \overline{ℂ} to be 0 on \overline{ℂ}−Λ(G) and to be equal to 1 on Λ(G). Contradiction. □

Note that the same proof works for geometrically finite subgroups of Isom(ℍ^{n}).

Conjecture 6.88 (Ahlfors measure zero conjecture.) Let G ⊂ Isom(ℍ^{n}) be a finitely generated discrete subgroup (n ≥ 3). Then the Lebesgue measure of Λ(G) is zero unless Λ(G) = ∂_{∞}ℍ^{n}.

This conjecture was first formulated by Ahlfors around 1966, however remains open even if n = 3. If G ⊂ Isom(ℍ^{3}) is freely indecomposable or ℍ^{3}/G admits a compactification to a manifold with boundary, then Ahlfors measure zero conjecture has positive solution, see [24, 34].

6.19 Supgroups of finitely generated Kleinian groups

Theorem 6.89 Let G ⊂ Isom(ℍ^{3}) be a finitely generated nonelementary Kleinian group; Isom(ℍ^{3}) ⊃ H ⊃ G, H is a group such that Λ(G) is invariant under H. If H is discrete then |H : G| < ∞. Assume that the limit set of G is not contained in a round circle. Then H is discrete.

Proof: Suppose that H is not discrete. Then the connected component of identity cl(H)^{0} in the closure cl(H) of H in Isom(ℍ^{3}) is either conjugate to P SL(2, ℝ) or equals Isom(ℍ^{3}). Since Λ(G) is H-invariant, it is also cl(H)^{0}-invariant. The only closed P SL(2, ℝ)-invariant subsets of \overline{ℂ} are ∅, \overline{ℂ} and a circle in \overline{ℂ}. Thus cl(H)^{0} = Isom(ℍ^{3}). However Isom(ℍ^{3}) does not have any proper invariant subsets in \overline{ℂ}, which shows that H is discrete. Moreover, Λ(H) = Λ(G) (since the limit set of H is the smallest nonempty H-invariant subset of \overline{ℂ}). Thus H is a Kleinian group. Consider the covering p : Ω(G)/G → Ω(H)/H induced by the inclusion G ⊂ H. According to Ahlfors finiteness theorem, the hyperbolic area of Ω(G)/G is finite. This implies that the covering p is finite, therefore |H : G| < ∞. □

6.20 Ends of hyperbolic manifolds

If you are familiar with the definition of an end of a topological space, you can disregard the following discussion of this definition. Otherwise, before reading the following definition, first read the definition of a a based filter in Section 10.1 .

First recall the general definition of ends of noncompact manifolds. Suppose that M is a topological manifold (possibly with boundary). Let S denote the collection of subsets S ⊂ M so that:

• each S is the interior of a nonempty submanifold S̄ ⊂ M which has compact boundary;

• S̄ has exactly one noncompact component.

Definition 6.90 A (topological) end of the manifold M is a filter E based on S, which satisfies the extra condition:

⋂_{U∈E} U = ∅

Any set U ∈ E is called a neighborhood of the end E. This looks like an awful definition, in practice however we will have to deal only with simple examples of ends, here is the one. Let N be a compact manifold with boundary, L ⊂ ∂N be a compact subsurface (it could be the entire boundary). Let M := N − L. Suppose L consists of connected components L_{1}, ..., L_{k}. Then ends of M are in 1-1 correspondence with components L_{j}. To construct a system of neighborhoods for each L_{j} take the system of compact codimension zero submanifolds with boundary N_{αj} ⊂ N each of which contains L_{j}, and let {U_{jα} := int(N_{αj})}. This collection forms a base of the corresponding filter E_{j}. Ends which appear this way are called topologically tame.

Suppose that M is a 3-manifold with the finitely-generated fundamental group, M^{c} is a compact core of M. Then there is 1-1 correspondence between the components N_{j} of M − M^{c} and the ends of M (each N_{j} is a neighborhood of one and only one end). Thus, by abusing notations, I will say that N_{j} is an end of M.

Figure 11:

Let M be a complete hyperbolic 3-manifold with the finitely generated fundamental group, for ϵ < μ_{3} let M_{ϵ}^{0} be the union of M_{[ϵ,∞)} and Margulis tubes in M_{(0,ϵ]}. The manifold with boundary M_{ϵ}^{0} has a Scott compact core M_{ϵ}^{c} defined in Theorem 6.81. We will call connected components of M_{[ϵ,∞)}−M_{ϵ}^{c} geometric ends of the manifold M. In what follows we shall refer to the geometric ends as simply ends. Thus, ends of M are in 1-1 correspondence with the boundary components of M_{ϵ}^{c} − ∂M_{ϵ}^{0}. The parabolic extension P (E) of the end E of M is the union of E and those cusps in M_{(0,ϵ]} which are adjacent to the boundary of E. Let ∂_{P} E denote the frontier of E in P (E) and ∂_{0}E := ∂E − ∂_{P} E, ∂_{0}M_{ϵ}^{c} is defined as the union of ∂_{0}E over all ends E of M_{ϵ}^{0}. See Figure 11.

Let M be a complete hyperbolic 3-manifold with the finitely-generated fundamental group G ⊂ Isom(ℍ^{3}). Let CM denote the convex core of M. An end E of M^{0} is said to be geometrically finite if E ∩ CM is compact. Another way to express this is to say that the subset E ⊂ M^{0} is precompact in the compactification M̄ = M ∪ Ω(G)/G. Note that G is geometrically finite if and only if all ends of M are geometrically finite.

Now consider the following special case. Take a complete hyperbolic surface Σ of finite area, which has genus g and m punctures. Consider a complete hyperbolic 3-manifold M so that Γ := π_{1}(M) ≅ π_{1}(Σ), where the isomorphism ρ : π_{1}(S) → π_{1}(M) is a parabolic representation. Thus, images of peripheral loops in Σ are parabolic elements in Γ, we call them regular parabolic elements of Γ. Let P := P_{1} ∪ ... ∪ P_{m} be the union of cusps in M which correspond to regular parabolic elements (we call them regular cusps). However, it could happen that some other (nonperipheral) elements γ in π_{1}(Σ) are also mapped to parabolic elements. Such elements ρ(γ) are called accidental parabolic elements. Let Q := Q_{1} ∪ ... ∪ Q_{n} denote the complete collection of Margulis cusps in M which correspond to accidental parabolic elements (we call them accidental cusps).

Our current goal is to understand the structure of accidental cusps in M. Any Scott compact core M_{c} of M is homeomorphic to Σ_{c} × [−1, 1], where Σ_{c} is a compact subsurface in Σ obtained by removing disjoint punctured disks around cusps. Theorem 6.82 implies that we can choose M_{c} so that:

M_{c} intersects the boundary of each cusp Q_{j}, P_{i} along a single annulus A_{j}, B_{i} which carries the fundamental group of Q_{j}, P_{i}.

The union M_{c} ∪ P splits the manifold M in two pieces: the “positive part” M^{+} ⊂ M − M_{c} ∪ P (adjacent to Σ_{c} × {+1}) and the “negative part” M^{−} = M − M_{c} ∪ P − M^{+}. Therefore all accidental cusps in M group in two collections: positive cusps Q_{j}^{+} (so that A_{j}^{+} ⊂ M^{+}) and negative cusps Q_{j}^{−} (so that A_{j}^{−} ⊂ M^{−}). The annuli A_{j}^{±} correspond to simple closed loops γ_{j}^{±} ⊂ Σ which are homotopic to A_{j}^{±}. The distinct loops γ_{j}^{+}, γ_{i}^{+} (and γ_{j}^{−}, γ_{i}^{−}) are disjoint. Thus we obtain two systems of simple disjoint loops Π = Π^{+} ∪ Π^{−} on Σ such that each accidental cusp in M^{±} is represented by a curve in Π^{±}.

Lemma 6.91 No two curves in collections Π^{+}, Π^{−} are homotopic.

Proof: Suppose that γ^{+} ∈ Π^{+} is homotopic to γ^{−} ∈ Π^{−}. Then γ^{±} represent cusps Q^{±} ⊂ M^{±}. The cusps Q^{±} are disjoint and their fundamental groups are conjugate in π_{1}(M). This contradicts Lemma 6.48 □

Corollary 6.92 The cardinality of the collection Π is at most 2(3g − 3 + m).

7 Introduction to orbifold theory

7.1 Definitions and examples

The notion of orbifold is a generalization of the notion of a manifold which appears naturally in the context of properly discontinuous non-free actions of groups on manifolds. Orbifolds were first invented by Satake [174] in 1950-s under the name of V-manifolds, they were reinvented under the name of orbifolds by Thurston in 1970-s, as a technical tool in proving the hyperbolization theorem. Our discussion of orbifolds mainly follows [192, Chapter 13] and [180].

Note that many definitions that we take for granted in the manifold theory are rather misterious in the category of orbifolds. For instance, it is unclear what is the correct definition of characteristic classes of orbifolds: even characteristic numbers are not welldefined (the differential-geometric definition given in [114] seems to be different from the definition which one would like to have for complex-algebraic orbifolds, the latter definition is supposed to behave well under resolution of singularities, see [91]). Even if one has the Euler characteristic of an orbifold it is unclear what the Betti numbers are, we will define the fundamental group of an orbifold but it is far from obvious what is the correct definition of higher homotopy groups, etc. All in all, it seems that the orbifolds live in the area between the “good old manifold kingdom” and the dark world populated by the wild creatures of the Alan Connes’ “noncommutative geometry”.

Before giving a formal definition we start with basic examples of orbifolds. Suppose that M is a smooth manifold and G is a discrete group acting smoothly, effectively and properly discontinuously on M. Then the quotient O = M/G is an orbifold, such orbifolds are called good. The quotient M/G, considered as a toplogical space X_{O}, is the underlying set of this orbifold. If S is a set of points in M where the action of G is not free, then its projection Σ = S/G is the singular locus of the orbifold O.

To be more concrete consider 2-dimensional orbifolds. Suppose that M = ℍ^{2} and G is a discrete subgroup of P SL(2, ℝ). Then the quotient O = ℍ^{2}/G is a Riemann surface X_{O} with a discrete collection of cone points z_{j} which form the singular locus Σ of the orbifold O. The projection p : ℍ^{2} → O is the universal cover of the orbifold O. The Riemann surface X_{O} has a natural hyperbolic metric which is singular in discrete set Σ. Metrically the points z_{j} are characterized by the property that the total angles around these points are 2π/n_{j}. The numbers n_{j} are the orders of cyclic subgroups G_{z_{j}} of G which stabilize the points in p^{−1}(z_{j}), they are called the local isotropy groups. The projection p is a ramified covering from the point of view of Riemann surfaces. From the point of view of orbifolds it is a covering. Thus the singular locus of the orbifold O consists of points z_{j} in Σ equipped with the extra data: the P SL(2, ℝ)-conjugacy classes of local isotropy groups G_{z_{j}} (of course each local isotropy group G_{z_{j}} is determined by the number n_{j}).

More complicated examples appear when G ⊂ Isom(ℍ^{2}) does not preserve orientation: such groups can contain isometric reflections. Then near each point of the quotient ℍ^{2}/G the orbifold O can look like a piece of closed half-plane in ℍ^{2}. The local structure of the orbifold near such points is called the boundary reflector. More generally, reflections R, L ∈ G can have a common fixed point z ∈ ℍ^{2}. In this case the group ⟨R, L⟩ is a finite dihedral group. The quotient of ℍ^{2} near the projection of such point z looks like a corner with geodesic sides in ℍ^{2}. The angle at this corner equals π/n, where n is the order of the element R · L. Another way to think about this local picture is to take a small metric disk D_{r}(z) with the center at z. Then the group ⟨R, L⟩ acting on D has a fundamental domain which is a corner with the vertex z and edges fixed by the elements R and L. The local structure of the orbifold near the point z is the corner reflector. See Figure 12.

Figure 12: 2-dimensional orbifolds.

Let G be a discrete subgroup of Isom(ℍ^{2}) generated by reflections with the fundamental domain D bounded by geodesics fixed by reflections. Then the orbifold O = ℍ^{2}/G is naturally identified with the domain D; the orbifold data is given by the metric structure of O, namely by the angles at the corners.

Now we can discuss the general definition. A (smooth) n-dimensional orbifold O is a pair: a Hausdorff paracompact topological space X (which is called the underlying space of O and is denoted X_{O} or |O|) and a special atlas A on X. The atlas A consists of:

• A collection of open sets U_{i} ⊂ X,which is closed under taking finite intersections, so that X = ⋃_{i} U_{i}.

• A collection of open sets Ũ_{i} ⊂ ℝ^{n}.

• A collection of finite groups of diffeomorphisms Γ_{j} acting on Ũ_{i}.

• A collection of homeomorphisms

ϕ_{i} : U_{i} → Ũ_{i}/G_{i}

We require the atlas A to behave well under inclusions. Namely, suppose that U_{i} ⊂ U_{j}, then there is a smooth embedding

ϕ̃_{ij} : Ũ_{i} → Ũ_{j}

and a monomorphism f_{ij} : G_{i} → G_{j} so that ϕ̃_{ij} is f_{ij}-equivariant.

We will call U_{j} coordinate neighborhoods of points x ∈ U_{j} and Ũ_{j} their covering coordinate neighborhoods .

Similarly to orbifolds we define the class of orbifolds with boundary; just instead of open sets Ũ_{j} ⊂ ℝ^{n} we use open subsets in

ℝ_{+}^{n} ∪ ℝ^{n−1} = {(x_{1}, ..., x_{n}) : x_{n} ≥ 0}

The boundary of such orbifold consists of points x ∈ |O| which correspond to ℝ^{n−1} under the identification U_{i} ≅ Ũ_{i}/G_{i}. As in the case of manifolds, the boundary of each orbifold is an orbifold without boundary. By abusing notations we will call an orbifold with boundary simply an orbifold.

An orbifold with empty singular locus is called nonsingular or a manifold.

To each point x ∈ X we associate a germ of action of a finite group of diffeomorphisms G_{x} at a fixed point x̃. If ϕ_{j}(x) is covered by a point x̃_{j} ∈ Ũ_{j}, then we have the isotropy group G_{j,x} of x̃_{j} in G_{j}. Note that if U_{i} ⊂ U_{j}, then the map ϕ̃_{ij} : Ũ_{i} → Ũ_{j} induces an isomorphism from the germ of the action of G_{j,x} at x̃_{j} to the germ of the action of G_{i,x} at x̃_{i}. Thus we let the germ (G_{x}, x̃) be the equivariant diffeomorphism class of the germ (G_{j,x}, x̃_{j}). The group G_{x} is called the local isotropy group of O at x. The set of points x with nontrivial local isotropy group is called the singular locus of O and is denoted by Σ_{O}. Note the the singular locus is nowhere dense in X_{O}.

Since G_{x} acts smoothly near the fixed point x̃, the germ (G_{x}, x̃) is linearizable: give a neighborhood of x̃ a G_{x}-invariant Riemannian metric. Then the exponential map (with the origin at x̃) conjugates the orthogonal action of G_{x} on the tangent space Tx̃ℝ^{n} to the germ of the action of G_{x} at x̃. Thus we can talk about reflections in the group G_{x} (they correspond to the reflections in O(n) under the above conjugations), order of rotation of an element g ∈ G_{x} if n ≤ 3 and g is not a reflection.

Definition 7.1 A Riemannian metric ρ on orbifold O is the usual Riemannian metric on X_{O} − Σ_{O}, so that after we lift ρ to local covering coordinate neighborhoods Ũ_{i}, it extends to a G_{i}-invariant Riemannian metric on the whole Ũ_{i}.

Exercise 7.2 Each orbifold O admits a Riemannian metric. Hint: use the partition of unity argument similar to the manifold case.

A homeomorphism (resp. diffeomorphism) between orbifolds O, R is a homeomorphism h : |O| → |R| so that for each point x ∈ O, y = h(x) ∈ R there are coordinate neighborhoods U_{x} ≅ Ũ_{x}/G_{x}, V_{y} ≅ Ṽ_{y}/G_{y} so that h lifts to an equivariant homeomorphism (resp. diffeomorphism)

h̃_{xy} : Ũ_{x} → Ṽ_{y}

Note that to describe a smooth orbifold O up to homeomorphism it suffices to describe topology of the pair (X_{O}, Σ_{O}) and the homeomorphic equivalence classes of the germs (G_{x}, x̃) for the points x ∈ Σ_{O}.

Example 7.3 Let O be a connected compact 1-dimensional orbifold without boundary which is not a manifold. Then O is homeomorphic to the closed interval [ab] where (G_{a}, ã), (G_{b}, b̃) are the germs (ℤ_{2}, 0) of the reflection group ℤ_{2} acting isometrically on ℝ near its fixed point 0 ∈ ℝ.

A smooth map between orbifolds O and R is a continuous map

g : O → R

which can be (locally) lifted to a map of pairs of coordinate covering neighborhoods

g̃_{ij} : Ũ_{j} → Ṽ_{i}

as a smooth equivariant map which induces isomorphisms of local isotropy groups G_{x} → G_{y}, x ∈ O, y = g(x) ∈ R. (Unfortunatly there is no standard definition of morphism between orbifolds in the literature, so we have chosen one which is convenient for our purposes.) Similarly we define immersions and submersions between orbifolds as smooth maps between orbifolds which locally lift to immersions and submersions respectively.

Example 7.4 Let O be the 2-dimensional reflection orbifold so that X_{O} is the closed unit disk Δ^{2}, Σ_{O} is the boundary circle of O and O has the local structure of boundary reflector near each singular point. Let D ⊂ Δ^{2} be a segment connecting two boundary points. Give D the orbifold structure as in the Example 7.3. Then the inclusion D ↪ O is an immersion. Let X_{D} be the manifold with boundary underlying D, treat X_{D} as orbifold with boundary with empty singular locus. Then the inclusion ι : X_{D} → O is not a smooth map. Let R be the orbifold whose underlying topological space X_{O} is the unit square in ℝ^{2}, the vertices of this square are corner reflectors, the interiors of the edges are boundary reflectors, Σ_{R} is the boundary of the square. Let j : D ↪ R be inclusion of the above orbifold D are a straight-line segment connecting the opposite vertices of the square. Then j is not a smooth map.

A suborbifold R ⊂ O is an orbifold R together with an embedding (i.e. an injective immersion) ι : R ↪ O. We say that a suborbifold R is properly embedded in the orbifold O if the embedding ι : R ↪ O sends ∂R to ∂O, ι^{−1}(∂O) ⊂ ∂R and written in local covering coordinates the map ι is transversal to ∂O.

Suppose that O′, O are orbifolds and p : |O′| → |O| is a continuous map. The map p is called a covering map between the orbifolds O′, O if the following property is satisfied:

For each point x ∈ |O| there exists a chart U = Ũ/G_{x} so that for every component V_{i} of p^{−1}(U) the restriction map p : V_{i} → U is a quotient map of an equivariant diffeomorphism h_{i} : Ṽ_{i} → Ũ (if y_{i} = p^{−1}(x) ∩ V_{i} then h_{i} conjugates the action of G_{y_{i}} on Ṽ_{i} to action of a subgroup of G_{x} on Ũ).

From now on we will assume that the orbifolds under consideration are connected.

The universal covering p : Õ → O of the orbifold O is the initial object in the category of orbifold coverings, i.e. it is a covering so that for any other covering p′ : O′ → O there exists a covering p̃ : Õ → O′ such that p′ ∘ p̃ = p. If p : Õ → O is the universal covering then the orbifold Õ is called the space universal covering space of O (by obusing notations we will also refer to Õ as the universal covering).

The group Deck(p) of deck transformations of the orbifold covering p : O′ → O is the group of self-diffeomorphisms h : O′ → O′ such that p ∘ h = p. A covering p : O′ → O is called regular if O′/Deck(p) = O.

The fundamental group π_{1}(O) of the orbifold O is the group of deck transformations of its universal covering. Then O = Õ/π_{1}(O).

The following theorem was proven by Thurston in [192, Chapter 13] (it will not be needed in these notes).

Theorem 7.5 Each orbifold has a universal covering.

Definition 7.6 An orbifold O is called good if its universal cover is a manifold. Orbifolds which are not good are called bad. An orbifold is called very good if is admits a finite manifold cover.

Theorem 7.7 (Armstrong Theorem, [9].) Suppose that O is a good orbifold. Then π_{1}(X_{O}) = π_{1}(O)/T, where T ⊂ π_{1}(O) is the normal subgroup generated by all elements which do not act freely on the universal cover Õ.

As in the case of manifolds, there is a form of Seifert-Van Kampen theorem which allows to compute fundamental groups of orbifolds, we leave its proof as an excercise to the reader. (Hint: use the proof of Seifert-Van Kampen theorem which is based on the group actions on trees, see [180, Section 2] where P. Scott discusses the 2-dimensional case.)

Theorem 7.8 Let O be a connected orbifold, O_{1}, O_{2} ⊂ O are open connected suborbifolds so that:

• O = O_{1} ∪ O_{2};

• O_{1} ∩ O_{2} is connected; • Closures of O_{1}, O_{2} are suborbifolds with boundary in O, so that the boundary of O_{i}

equals the frontier of X_{O_{i}} in X_{O}.

Let ı_{i} : O_{i} ↪ O denote the inclusion. Then the fundamental group of O is obtained from π_{1}(O_{1}) ∗ π_{1}(O_{2}) by adding the relations γ_{1}γ_{2}^{−1} = 1 where γ_{i} ∈ π_{1}(O_{i}), i = 1, 2, γ_{i} = ı_{i∗}(γ), i = 1, 2, for each γ ∈ π_{1}(O_{1}) ∩ π_{1}(O_{2}).

It is easy to see that for any compact orbifold O the fundamental group π_{1}(O) is finitely generated. To prove this use either Seifert-Van Kampen theorem for orbifolds or 4.22. (Actually π_{1}(O) is finitely presentable.)

The groups G_{i} and transition maps ϕ̃_{ij} which appear in the definition of smooth orbifolds may have more restrictive nature: we may require that (after restriction to open subsets) they belong to a Lie group H acting isometrically on a Riemannian manifold M. Then O is called a geometric orbifold modelled on the geometry (M, H). The example that we will be interested in is M = ℍ^{n}; H = Isom(ℍ^{n}), in which case O is a hyperbolic orbifold. More generally, define (locally) CAT (k)-orbifold by taking a complete CAT (k)-manifold M as the model space and H as the (pseudo)-group of isometries of M.

An orbifold O is called locally reflective if the local isotropy group at each point is generated by reflections. For instance, 2-dimensional locally reflective orbifolds locally near each singular point look like either boundary reflectors or corner reflectors.

Any geometric orbifold is a Riemannian manifold outside of the singular locus, the geometric structure provides the whole space X_{O} with a path-metric d_{O} defined exactly like in the Riemannian case. The geometric orbifold O is complete if (X_{O}, d_{O}) is complete as a metric space. The same definition applies to locally CAT (k)-orbifolds.

Suppose that O is a geometric orbifold modelled on (M, H). Then there is a well-defined (up to conjugation in H) homomorphism ρ : π_{1}(O) → H which is called the holonomy representation of O and the group ρ(π_{1}(O)) is the holonomy group of O.

Proposition 7.9 Suppose that O is a complete hyperbolic orbifold without boundary so that π_{1}(O) is a finitely generated group. Then O is very good.

Proof: We give here only a sketch of the proof, see [137] for details. As in the case of manifolds modelled on the geometry (ℍ^{n}, Isom(ℍ^{n})) we can define the developing map dev : Õ → ℍ^{n} on the universal cover of O. As in the Riemannian case, completeness of O implies that dev is a diffeomorphism. Thus O = ℍ^{n}/G where G is the holonomy group of O. The proposition now directly follows from the Selberg Lemma. □

Remark 7.10 To guarantee that O is a good orbifold much weaker condition suffices: if O is a complete (locally) CAT (0)-orbifold then O is good; this is a version of Cartan-Hadamard theorem due to Haefliger, see [85].

7.2 2-dimensional orbifolds

Recall that according to our definition, the local isotropy group G_{x} of each 2-dimensional orbifold O can be conjugate (via the exponential map) to a finite group of isometries G_{x} ⊂ Isom(ℝ^{2} ≅ ℂ) which fixes the origin. Thus we have the following possibilities for the group G_{x}:

(1) G_{x} = {1};

(2) G_{x} ≅ ℤ_{n} is generated by rotation of order n ≥ 2, the number n is called the order of the singular point x; (3) G_{x} is conjugate to the group ℤ_{2} generated by the reflection z ↦ z̄;

(4) G_{x} is the dihedral group Dih_{m} generated by two reflections τ, σ so that (τσ)^{m} = 1.

If only the first two possibilities occur then O is called orbifold of cone type. Such orbifolds can be described combinatorially as follows. Start with a surface S (possibly with boundary), let P ⊂ S be a discrete subset disjoint from the boundary of S. Mark each point p ∈ P with an integer n_{p} ≥ 2. Then the cone orbifold structure associated with this data is: G_{x} = {1} for each x ∈ S − P, G_{p} ≅ ℤ_{n_{p}} for each p ∈ P. Using Seifert-Van Kampen theorem the fundamental group of such orbifold O (if it is connected) can be computed as follows (see [180] for an alternative computation):

Take the fundamental group Γ of the surface S − P. For each peripheral element γ_{p} ∈ Γ corresponding to a simple loop in S − P which goes once around the puncture p ∈ S we add the relation γ^{n_{p}} = 1 to the group Γ. The result is isomorphic to π_{1}(O).

Equivalently, let surface with boundary S̄ be the (partial) compactification of S − P obtained by adding a circle γ_{p} for each puncture p ∈ P. Then along each boundary loop γ_{p} of S̄ add a 2-cell so that the attaching map has degree ±n_{p}. The fundamental group of the resulting CW-complex is isomorphic to π_{1}(O).

Figure 13: Bad 2-dimensional orbifolds.

Here is the complete list of bad 2-dimensional orbifolds (see Figure 13):

1. The teardrop: the underlying set of O is the 2-sphere and Σ_{O} is a single point.

2. The double teardrop: the underlying set of O is the 2-sphere and Σ_{O} is a pair of conical points with distinct orders.

3. The quotient of the teardrop by the reflection in a circle containing the singular point.

4. The quotient of the double teardrop by the reflection in a circle containing both singular points.

Any good 2-dimensional orbifold (with finitely generated fundamental group) is also very good. There are 3 disjoint classes of good compact 2-dimensional orbifolds without boundary:

• Spherical orbifolds, which are quotients of the round 2-sphere by a finite group of isometries.

• Euclidean orbifolds, which are quotients of the Euclidean plane by lattices.

• Hyperbolic orbifolds, which are quotients of the hyperbolic plane by uniform lattices.

See [180] for details.

To get a 2-dimensional compact orbifold with boundary take a 2-dimensional closed orbifold O and remove from its singular locus a finite collection of open disjoint arcs lying on ∂X_{0} and/or a collection of open disks from X_{O} − Σ_{O}.

Exercise 7.11 Any compact 2-dimensional orbifold can be obtained this way.

Figure 14: Orbifolds covered by D^{2}.

Now we will give lists of 2-dimensional orbifolds which are finitely covered by disks, annuli and tori. Suppose that O is a compact 2-orbifold which admits a manifold cover homeomorphic to the compact disk D^{2}. Here is the complete list of possibilities for the orbifold O (see Figure 14):

• O ≅ D^{2} and Σ_{O} = ∅.

• |O| = D^{2} and O has a single singular cone point (of order n > 1), thus O is the quotient of D^{2} by a group of rotations of order n.

• O is the quotient of D^{2} by a single reflection, the singular set of O consists of a single boundary reflector arc.

• O is the quotient of D^{2} by a dihedral group Dih_{n}. Thus |O| = D^{2} and O has one corner reflector singular point, the rest of the singular set consists of two edges which are boundary reflectors.

Suppose that O is a compact 2-orbifold which admits a manifold cover homeomorphic to the annulus A = 𝕊^{1} × I and O ≠ A, (see Figure 15). The either:

• O ≅ μ is the Moebius band.

• |O| = A and one boundary component of A is the boundary reflector, we call this orbifold the 1-edged annulus.

Figure 15: Orbifolds covered by the annulus 𝕊^{1} × I.

• |O| is the rectangle, two disjoint edges of this rectangle are boundary reflectors, O is the quotient of A by a single reflection; we will call this orbifold the 2-edged square.

• |O| = D^{2} which has two singular cone points of order 2, thus O is the quotient of A by a group of rotations of order 2.

• O is the quotient of A by the dihedral group ℤ/2 × ℤ/2, |O| is the rectangle, three edges of this rectangle are boundary reflectors. We will call this orbifold the 3-edged square.

Finally consider orbifolds covered by the torus. We shall assume that these are locally reflective orbifolds.

• |O| is the rectangle, it has 4 corner reflectors and 4 edges which are boundary reflectors; we will call this orbifold the mirror square.

• |O| is a triangle, it has 3 corner reflectors and 3 edges which are boundary reflectors; possible angles are: (π/2, π/4, π/4), (π/3, π/3, π/3), (π/2, π/3, π/6); we will call these orbifolds Euclidian triangle orbifolds.

• |O| is the annulus, both of its boundary components are boundary reflectors; we will call this orbifold the mirror annulus.

• |O| is the Moebius band, its boundary is the boundary reflector; we will call this orbifold the mirror Moebius band.

7.3 3-dimensional locally reflective orbifolds

The class of 3-dimensional orbifolds that we will use, consists of locally reflective orbifolds. Underlying spaces of these orbifolds are 3-manifolds (with boundary). Below we give a combinatorial description of these orbifolds. Suppose that X is a compact 3-manifold with boundary, Γ̂ ⊂ ∂X is a 3-valent graph. We introduce structure of a locally reflective orbifold O with the underlying set X as follows. Pick a collection of components of ∂X − Γ̂, its closure S will be the designated boundary ∂O. In what follows we will treat S as an orbifold, so we shall use the notation |S| to denote the underlying surface with boundary. Let Σ := ∂X − int(|S|), the designated singular locus. For each point x ∈ X − Σ the local isotropy group G_{x} is trivial. Let Γ := cl(Γ̂−|S|). We count as vertices of Γ only the vertices of valency 3, the rest of vertices are dead ends which belong to ∂|S|. Choose a function

θ : edges of Γ → {π/2, π/3, ..., π/n, ...}

Assume that for any vertex v of Γ and edges e_{1}, e_{2}, e_{3} emanating from v satisfy the condition:

θ(e_{1}) + θ(e_{2}) + θ(e_{3}) > π

(See Figure 16.)

Components of Σ − Γ̂ are faces of the orbifold O. For each point x in a face F, the local isotropy group G_{x} is isomorphic to ℤ_{2} and the generator acts as a reflection whose fixed-point set projects to F. Edges of the graph Γ are called edges of the orbifold O. If x belongs to an edge e of O but is not a vertex, then its local isotropy group G_{x} is a dihedral group Dih_{m} = ℤ_{2} ⋉ ℤ_{m} where 1/m = θ(e)/π. Theis group is generated by two isometric reflections in ℝ^{3} so that the intersection of their fixed-point sets projects (locally) to e. Finally, if x is a vertex of Γ, then G_{x} is a finite Coxeter subgroup of O(3) generated by three reflections which has the presentation:

⟨τ_{1}, τ_{2}, τ_{3} : (τ_{j})^{2} = 1, (τ_{i}τ_{j})^{m_{ij}} = 1⟩

where m_{ij}^{−1} = θ(e_{ij})/π and e_{ij} is the edge of Γ between the faces corresponding to τ_{i}, τ_{j}. The group G_{x} is a spherical triangle group.

Figure 16: Combinatorial structure of a locally reflective orbifold.

Proposition 7.12 Any locally reflective compact 3-dimensional orbifold O can be obtained as above.

Proof: The underlying space X = X_{O} is a compact manifold with boundary. Since O is locally reflective, the singular set Σ of O is contained in ∂X. The local isotropy group G_{x} of each point is a finite subgroup of O(3) generated by reflection. The list of such groups

Figure 17: Examples of the parabolic locus of a locally reflective orbifold.

is very small, it consists of ℤ/2, dihedral groups Dih_{m} and spherical triangle groups above. According to these local isotropy groups we get a stratification of Σ into faces, edges and vertices. Complement to the union of faces in Σ is the graph Γ̂. □

Note that S = ∂O inherits a locally reflective orbifold structure. The corresponding 2dimensional orbifold has the underlying space |S|, its singular set consists of the boundary reflectors which are edges of Γ̂ ∩ ∂S, besides there are corner reflectors corresponding to the end-points of these edges.

In what follows we will need to specify the parabolic locus on the boundary of a locally reflective orbifold, similarly to the description of the parabolic locus on the boundary of a compact 3-manifold. First,collect all components of ∂O which have the structure of Euclidean orbifolds, then we pick a collection of disjoint Euclidean suborbifolds with boundary which are contained in ∂O (in the interesting case these components are nonsingular annuli and 2-edged and 3-edged squares.) The total collection of these suborbifolds P = P_{1}∪...∪P_{n} is the parabolic locus of O. We let S^{0} := cl(S − P). Later on we shall impose some further restrictions on these components which generalize the definition of a paired 3-manifold. See Figure 17.

Define the orbifold structure on faces of locally reflective orbifolds O as follows. Each face F of O has a natural polygonal structure, where edges appear as intersections of F with other faces of O as well as with the boundary of O. We declare the former arcs to be boundary reflectors of the orbifold O_{F} whose boundary is F ∩ ∂O and the underlying space is F. To define the angle of the corner reflector of O_{F} at the vertex v (between two boundary reflectors) we use the “dihedral” angle of O at the edge of Γ emanating from v. See Figure 18.

Definition 7.13 Suppose that O is a locally reflective orbifold. An element g ∈ π_{1}(O) is called a reflection if it (locally) acts as a reflection in Õ. The face of this reflection is the projection of the fixed-point set of g to O.

Figure 18: Orbifold structure on faces of a locally reflective orbifold.

Theorem 7.14 (J. Morgan, [150], M. Kato [113].) Each compact locally reflective 3orbifold which contains no bad 2-suborbifolds is very good.

7.4 Orbifolds of finite type

In the category of 3-dimensional orbifolds one can define generalizations of compressing disks, essential annuli, irreducible, boundary irreducible, acylindrical and atoroidal orbifolds, incompressible surfaces etc. These definitions are straightforward generalizations of corresponding definitions in the class of manifolds and we collect them in the following Glossary which I in part had borrowed from [49].

GLOSSARY

Abad 3-orbifold: a 3-orbifold which contains no bad 2-suborbifolds.

Acylindrical 3-orbifold: a 3-orbifold where each properly embedded ANNULUS A has the property: ∂A bounds an ANNULUS in ∂O.

ANNULUS: a 2-orbifold A which is finitely covered by an annulus.

Topologically atoroidal 3-orbifold: a 3-orbifold O with the following properties:

• ∂O is incompressible, no boundary component is a 2-SPHERE;

• each incompressible TORUS in O is boundary parallel;

• O is not finitely covered by the product of 2-torus and interval.

Bad n-orbifold: an orbifold whose universal cover is not a manifold.

BALL: a 3-orbifold which is homeomorphic to B^{3}/Γ, where Γ ⊂ O(3) is a finite group.

Boundary parallel (or “inessential”) 2-suborbifold S in a 3-orbifold O: a properly embedded suborbifold such that a component of O − S is homeomorphic to (0, 1] × S.

Boundary compressing DISK D for a 2-suborbifold S in a 3-orbifold O: it is a DISK such that: ∂D is a union of two arcs α ⊂ S, β ⊂ ∂O which satisfy the following property:

there is no arc γ ⊂ ∂O ∩ S so that γ ∪ α is the boundary of a DISK in S. Closed n-orbifold: a compact orbifold with empty boundary. CIRCLE: a connected 1-dimensional closed orbifold. Compressing ANNULUS A for a 2-suborbifold S ⊂ O^{3}: ∂A = α ∪ β is a disjoint

union of two CIRCLES, so that:

• α = A ∩ S, β ⊂ ∂O^{3};

• α does not bound a DISK in O^{3};

• α ∪ β does not bound an ANNULUS in S.

Compressing DISK D for a 2-suborbifold S in a 3-orbifold O, is a DISK such that D ∩ S = ∂D does not bound a DISK in S.

DISK: a 2-orbifold which is diffeomorphic to D^{2}/Γ, where Γ ⊂ O(2) is a finite group. Good orbifold: an orbifold which is covered by a manifold. Incompressible 2-suborbifold S of a 3-orbifold O: a suborbifold S which either has

no compressing DISKS and boundary compressing DISKS or S is a DISK whose boundary ∂S ⊂ ∂O does not bound any DISKS in ∂O.

Irreducible 3-orbifold: a 3-orbifold O where any SPHERE bounds a BALL. Properly embedded 2-suborbifold S in a 3-orbifold O: a suborbifold such that

∂S ∩ ∂O = ∂S and this intersection is transversal.

Seifert 3-orbifold: a good 3-orbifold which is finitely covered by a Seifert manifold. (Note that there is also a class of bad Seifert orbifolds, see [26]. For example, it is natural to consider the product of 𝕊^{1} and a bad 2-orbifold as a Seifert-fibered orbifold. However we will deal only with good 3-orbifolds.)

2-SPHERE: a 2-orbifold which is finitely covered by 𝕊^{2}. Superincompressible 2-suborbifold of a 3-orbifold: a 2-sided properly embedded

incompressible 2-suborbifold which has no compressing annuli.

TORUS: a 2-orbifold covered by the 2-torus. Very good orbifold: admits a finite manifold covering. Waldhausen orbifold: a compact 3-dimensional orbifold which is irreducible, abad

and contains an incompressible suborbifold.

Note that the we call Waldhausen orbifold an object similar to a Haken 3-manifold. However in general Waldhausen orbifolds do not admit a Haken hierarchy, see [49].

Now we give several alternative definitions:

Definition 7.15 A very good 3-dimensional orbifold O is called irreducible, (resp. acylindrical) if some (any) of its finite manifold covers is irreducible, (resp. weakly acylindrical). A very good 3-dimensional non-Seifert orbifold O is called topologically atoroidal, if some (any) of its finite manifold coverings is atoroidal.

Example 7.16 Here is an example of a compressible Euclidean suborbifold Q ⊂ O. Take the mirror square Q, which bounds a solid wedge, see Figure 19.

Figure 19: Solid wedge bounded by the mirror square: an example of compressible Euclidean orbifold.

Equivalence of the definitions from GLOSSARY and the Definition 7.15 follows from the following theorems:

Theorem 7.17 (Equivariant loop theorem.) Suppose that M is a compact manifold with boundary, Q ⊂ ∂M is a subsurface which is compressible in M. Let F be a finite group of diffeomorphisms acting on (M, Q). Then there exists a compressing disk (D^{2}, ∂D^{2}) ⊂ (M, Q) which is precisely invariant under its stabilizer in F.

Theorem 7.18 (Equivariant sphere theorem.) Suppose that M is a compact orientable manifold, Σ ⊂ M is a sphere which does not bound a ball in M. Let F be a finite group of diffeomorphisms acting on M. Then there exists a sphere 𝕊^{2} ⊂ M (or a projective plane) which does not bound a ball and is precisely invariant under its stabilizer in F.

Theorem 7.19 (Equivariant annulus theorem.) Suppose that M is a compact manifold with boundary, Q ⊂ ∂M is a subsurface such that the manifold (M, M − Q) is not acylindrical. Let F be a finite group of diffeomorphisms acting on (M, Q). Then there exists an essential annulus (or Moebius band) (A, ∂A) ⊂ (M, M − Q) which is precisely invariant under its stabilizer in F.

Theorem 7.20 (Equivariant torus theorem.) Suppose that M is a compact 3-manifold which is not Seifert, and not atoroidal. Let F be a finite group of diffeomorphisms acting on M. Then there exists an essential torus (or Klein bottle) T ⊂ M which is precisely invariant under its stabilizer in F.

Equivariant Loop and Sphere theorems were first proven using minimal surfaces in [143]. A purely combinatorial proof appeared later in [99]. Equivariant Torus theorem follows from uniqueness of the canonical JSJ decomposition of Haken manifolds into atoroidal and Seifert components. Similarly the Equivariant Annulus theorem follows from uniqueness of the characteristic submanifold in a Haken manifold (see also [99]).

The class of orbifolds that we will be mainly interested in consists of all right locally reflective 3-dimensional orbifolds, which we describe below. Suppose that O is a compact locally reflective 3-dimensional orbifold whose underlying space is orientable, S = ∂O and the graphs Γ, Γ̂ ⊂ ∂X_{O} and the function θ are as in Section 7.3. Choose a designated parabolic locus P of the orbifold O to be a collection of disjoint incompressible Euclidean suborbifolds P_{i} ⊂ S so that:

Axiom 1: The function θ is identically equal to π/2 and no component of P is a 1-edged annuli. Thus each component P_{i} ⊂ P is either:

(a) an incompressible TORUS T_{i} ⊂ S;

(b) a homotopically nontrivial (nonsingular) annulus A_{i} ⊂ S;

(c) a 2-edged square;

(d) a 3-edged square.

Figure 20: An example of the finite type orbifold O with the underlying space B^{3}. F(P^{(1)}) = F_{1}∪F_{21}∪P^{(1)}. Note that the orbifold O itself is reducible. Black boxes on this figure denote components of the parabolic locus, one of them is 2-edged square. F_{j}’s are the faces of O.

We let P^{(1)} be the union of all components of P which are 2-edged squares and F(P^{(1)}) be the union of P^{(1)} and of all faces of O which are adjacent to P^{(1)}.

Axiom 2: For any pair of distinct connected components P_{i}, P_{j} ⊂ P, the conjugacy classes of π_{1}(P_{i}), π_{1}(P_{j}) in π_{1}(O) intersect by a finite subgroup. The suborbifold S^{0} := S−P is incompressible in O.

Axiom 3: The graph Γ has no vertices of valency ≥ 2, only dead ends (which belong to ∂S).

Axiom 4: O is very good; the orbifold O − F(P^{(1)}) is irreducible, atoroidal and acylindrical.

Definition 7.21 We say that a compact paired 3-orbifold (O, P) is all right if it satisfies Axioms 1– 4.

Axiom 5: Suppose that λ ⊂ ∂X_{O} − F(P^{(1)}) is an arc with the end-points on ∂S^{0} which is transversal to Γ and which is not homotopic into S^{0} relatively to ∂λ. Then int(λ) intersects at least 3 edges of Γ.

Definition 7.22 We say that a compact 3-orbifold O has finite type if it satisfies Axioms 1– 5.

Remark 7.23 There is nothing special about this class of orbifolds, it is just what we need for the induction step in Thurston’s Hyperbolization Theorem. Figure 20 gives example of an orbifold of finite type.

Figure 21: Possible positions for the disk D.

Suppose that I is an arc in ∂X_{O} − F(P^{(1)}) with the end-points on the same component of ∂S^{0}, I is transversal to Γ and is homotopic into S^{0} relatively to ∂I. Then we connect the end-points of I in S^{0} by an embedded arc J ⊂ S^{0}. The union I ∪I′ bounds a disk D ⊂ ∂X_{O}. Assume that I intersects at most 4 edges of Γ. Since the orbifold O − F(P^{(1)}) is supposed to be irreducible and acylindrical, we conclude that I ∪I′ bounds a properly embedded disk B ⊂ ∂X_{O} and this disk satisfies one of the following combinatorial possibilities described in the Figure 21:

(i) The disk B contains exactly two vertex of Γ̂.

(ii) The disk B contains no vertices of Γ̂.

Definition 7.24 An “all right” orbifold O belongs to the class R if it satisfies:

Axiom 6: S^{0} = ∅, i.e. the whole boundary of O consists of incompressible tori, Klein bottles, mirror annuli, mirror Moebius bands and mirror squares.

Note that in this case P^{(1)} = ∅, hence the orbifold O itself is irreducible, atoroidal and acylindrical. If O is an orbifold of the class R, then we say that peripheral elements of π_{1}(O) are those which are conjugate into π_{1}(P_{j}) for some P_{j} ⊂ ∂O.

Later on we shall need another class of orbifolds as well. Suppose that O is an all right orbifold. O is said to be all right alternated orbifold if it satisfies

Axiom 7: Faces of O can be colored in white and red colors so that no faces of the same color are adjacent. The coloring of faces of O induces a coloring of the edges of S^{0} ⊂ ∂O. We require all edges of S^{0} to have the same color.

7.5 A homeomorphism theorem for a class of 3-orbifolds

In this section we will prove a homeomorphism theorem for orbifolds of the class R. We start by collecting some basic facts about orbifolds of this class.

Lemma 7.25 Suppose that O is an orbifold which belongs to the class R. Then for any reflection r ∈ π_{1}(O) the centralizer of r in π_{1}(O) contains ℤ ∗ ℤ.

Proof: Since O is irreducible, the face of the reflection r in O is an incompressible surface in |O|. This face F is neither euclidean nor spherical 2-orbifold since O is assumed to be atoroidal and irreducible. Thus the lift of F to the universal cover of O is simply-connected and its stabilizer in π_{1}(O) is isomorphic to a nonelementary discrete subgroup of Isom(ℍ^{2}). However, this stabilizer is isomorphic to the centralizer of r in π_{1}(O). □

Lemma 7.26 Suppose that O is an orbifold which belongs to the class R and Q is a very good compact irreducible 3-orbifold with the fundamental group isomorphic to π_{1}(O). Then Q also belongs to the class R.

Proof: Let ψ : π_{1}(O) → π_{1}(Q) be the isomorphism. The only assertion which has to be proven is that for any reflection r ∈ π_{1}(O) the image ψ(r) is again a reflection. Note that centralizer of the element ψ(r) ∈ π_{1}(Q) keeps the fixed-point set F ix(ψ(r)) invariant. If ψ(r) is not a reflection, its fixed-point set has dimension ≤ 1, which implies that the centralizer of ψ(r) is almost cyclic. This contradicts Lemma 7.25. □

We need to establish some properties of orbifolds of the class R. Recall that any such orbifold O is good, thus its universal covering space Õ is a manifold. We denote by Σ̃_{O} the lift of the singular set of O to Õ. Each component C of Õ−Σ̃_{O} is bounded by lifts of faces of the orbifold O. The boundary of C has a natural polyhedral structure. We call a connected nonempty smooth subsurface F ⊂ ∂ÕC to be a face of C if there is a reflection r ∈ π_{1}(O) such that F is a component of intersection between clC and the fixed-point set of r. Let F̂ denote the whole fixed-point set of the reflection r which corresponds to F. The sets F̂ are simply-connected, properly embedded and noncompact since O is irreducible. Thus we shall refer to these sets as planes in Σ̃_{O}. Note that the planes in Σ̃_{O} can have nonempty boundary. Choose a Reimannian metric on O and lift it to the usual Riemannian metric on Õ.

Lemma 7.27 Suppose that C is a component of Õ − Σ̃_{O} and F_{1}, F_{2} are disjoint faces of C. Then F̂_{1} ∩ F̂_{2} = ∅.

Proof: Let r_{j} denote the reflection in π_{1}(O) with the fixed-point set F̂_{j}. Choose points x_{j} ∈ F_{j}. Suppose that x ∈ F̂_{1} ∩ F̂_{2}. Then there exists a reflection r ∈ π_{1}(O) so that the fixed-point set F̂ of r separates x from C, thus it also separates x from the both points x_{1}, x_{2}. Therefore F̂ must intersect the sets F̂_{1}, F̂_{2}. Since O ∈ R, either F̂_{1} = F̂_{2} or they intersect by the right angle.

In the first case we get contradiction with irreducibility of O, in the second case we get a subgroup in π_{1}(O) generated by r, r_{1}, r_{2} which corresponds to a spherical triangle orbifold T (2, 2, 2). This is impossible by the assumption that O ∈ R. □

Introduce the following equivalence relation: two reflections r′, r′′ ∈ Γ = π_{1}(O) are called connected if their fixed-point sets belong to the same connected component of Σ̃_{O}. This equivalence relation can be characterized algebraically as follows. Reflections r′, r′′ are connected iff there exists a chain of reflections r_{1} = r′, r_{2}, ..., r_{n} = r′′ so that each pair r_{i}, r_{i+1} generates the dihedral group ℤ_{2} × ℤ_{2}.

We define a simplicial tree T = T_{O} as follows. Blue vertices of T are components of Õ − Σ̃_{O}, black vertices are components of Σ̃_{O}. Blue and black vertices are connected by an edge iff they are adjacent in Õ. The group Γ acts naturally on the tree T without inversions on edges.

Lemma 7.28 Suppose that g ∈ π_{1}(O) does not stabilize any black vertex of the tree T = T_{O}. Then for any reflection r ∈ π_{1}(O) the elements r, r′ := grg^{−1} are not connected.

Proof: Let F be the fixed-point set of r, F′ = g(F) is the fixed-point set of r′. If the elements r, r′ would be equivalent then their fixed-point sets F, F′ would be in the same component of Σ̃_{O}. □

The basic fact about 3-dimensional orbifolds that we will need is the following generalization of Waldhausen-Johannson theorem:

Theorem 7.29 Suppose that 3-orbifolds O, Q belong to the class R and their fundamental groups are isomorphic. Then O is homeomorphic to Q.

Remark 7.30 One can weaken the assumption that O, Q belong to the class R, however the above theorem will suffice for our purposes.

Proof: Let Γ = π_{1}(O), Δ := π_{1}(Q) and ψ : Γ → Δ be the isomorphism. First we note that ψ carries peripheral elements of Γ to peripheral elements of Δ (since O, Q have finite manifold covers, which are atoroidal, acylindrical and have incompressible boundary).

Let T = T_{O}, S = T_{Q} denote the simplicial trees corresponding to the decompositions of Õ, Q̃ into components, as described above. We know that the isomorphism ψ carries reflections in the group Γ to reflections in the group Δ (see the proof of Lemma 7.26). The equivalence relation for these reflections is obviously preserved by the isomorphism. Thus, by Lemma 7.28, stabilizers of black vertices of T are mapped to stabilizers of black vertices of S. Our goal is to prove the same for blue vertices.

Proposition 7.31 Suppose that r′, r′′ are connected reflections in Γ so that their fixedpoint sets F̂′, F̂′′ correspond to faces F′, F′′ of a component C of Õ − Σ̃_{O}. Then the same is true for the reflections ψ(r′), ψ(r′′).

Proof: The proof is by induction on the combinatorial length of the chain of reflections connecting r′, r′′. First suppose r′, r′′ are neighbors, i.e. that F̂′ ∩F̂′′ ≠ ∅ and r′ ≠ r′′. Then obviously the fixed-point sets F ix(ψ(r′)) and F ix(ψ(r′′)) intersect in Q̃ by the right angle and we can take any component of Q̃ − Σ̃_{Q}, which is adjacent to F ix(ψ(r′)) ∩ F ix(ψ(r′′)).

The next case is when there is a neighboring face F of C between F′ and F′′. Let F̂ = F ix(r), where r is a reflection in Γ. If a plane F̂′′′ = F ix(r′′′) intersects F̂ transversally, then the reflection r′′′ belongs to the stabilizer Stab(F̂) of F̂ in the group Γ. The same is true for the fixed-point sets in the manifold Q̃. The isomorphism ψ carries G = Stab(F̂) into ψ(G) = Stab(F ixψ(r)). We claim that this isomorphism is induced by a homeomorphism F̂ → F ix(ψ(r)). Indeed, by compactness of O and Q, the groups G, ψ(G) act cocompactly on the planes F̂, F ix(ψ(r)). Peripheral elements of G, ψ(G) are also peripheral elements of π_{1}(O), π_{1}(Q). Therefore by Theorem 5.6 there exists a homeomorphism F̂ → F ix(ψ(r)) which induces the isomorphism G → ψ(G).

As an intermediate corollary we have the following assertion that will be used later on:

Corollary 7.32 The orbifolds F ix(r)/Stab(F ix(r)) and F ix(ψ(r))/Stab(F ix(ψ(r))) are homeomorphic for each reflection r ∈ Γ.

Now we go back to the planes F̂′, F̂′′. They are disjoint, intersect F and they are not separated by any other plane in Σ̃_{O}. If a plane F ix(ψ(r′′′)) separates F ix(ψ(r′)) from F ix(ψ(r′′)), then F ix(ψ(r′′′)) must intersect the plane F ix(ψ(r)) and we get contradiction with existence of an equivariant homeomorphism F ix(r) → F ix(ψ(r)). Thus there exists a component B = ψ_{∗}(C) of Q̃ − Σ̃_{Q} so that the planes F ix(ψ(r′)), F ix(ψ(r)) and F ix(ψ(r′′)) correspond to consecutive faces of B. The general case follows by induction. □

Corollary 7.33 Suppose that an element g ∈ Γ stabilizes an edge of the tree T. Then ψ(g) will also stabilize an edge of the tree S.

Proposition 7.34 Suppose that g ∈ Γ fixes a blue vertex of the tree T.Then ψ(g) fixes a blue vertex of the tree S.

Proof: We already know that stabilizers of edges and black vertices in T, S are preserved under the isomorphism ψ. Hence if ψ(g) does not fix any blue vertex of S, it acts on S as a hyperbolic element (Lemma 11.2). Thus there exists an axis A for the action of ψ(g) on S. Let v be any vertex on A, then the points v, ψ(g)v are separated on S by a black vertex w. Choose a point x ∈ Q̃ within the component corresponding to the vertex v, there exists a plane F̂ ⊂ w ⊂ Σ̃_{Q} which separates x from ψ(g)x. We will treat F̂ = F ix(ψ(r)) as a locally finite relative cycle in Z_{2}^{lf}(Q̃, ∂Q̃; ℤ_{2}). Let [x, ψ(g)x] be a smooth arc in Q̃ obtained by lifting the geodesic arc [v, ψ(g)v] ⊂ S. Then the orbit Λ := ⟨ψ(g)⟩([x, ψ(g)x]) belongs to Z_{1}^{lf}(Q̃; ℤ_{2}) and the ℤ_{2} intersection number between F̂ and Λ equals 1.

Recall that the element g was stabilizing a component C of Õ − Σ̃_{O}, choose a properly embedded real line Λ′ ⊂ C invariant under g. Then Λ′ is disjoint from any plane F̂′ ⊂ Σ̃_{O}. This implies that the algebraic inteersection number between the locally finite cycles Λ′ ∈ Z_{1}^{lf}(Õ; ℤ_{2}) and F̂′ ∈ Z_{2}^{lf}(Õ, ∂Õ; ℤ_{2}) equals zero.

Now we use the fact that both orbifolds O, Q are very good. They admit a common finite regular manifold covering M (by Johannson’s theorem) and we shall identify Õ and Q̃ with the universal covering M̃ of M. The actions of Γ and Δ on M̃ descend to homotopic finite group actions on M. Thus there exists a proper ψ^{−1}-equivaraint homotopy-equivalence h : (M̃, ∂M) → (M̃, ∂M). The mapping h induces isomorphisms of the locally finite homology groups. Moreover, by equivariance, we have: [h(F̂)] = [F̂′] ∈ Z_{2}^{lf}(Õ, ∂Õ) for the plane F̂′ = F ix(ψ^{−1}(r)) and [h(Λ)] = [Λ′] in H_{1}^{lf}(Õ). Thus the algebraic intersection numbers between Λ, F̂ and Λ′, F̂′ must be equal. On the other hand, one of them is 1 and the second is zero. Contradiction. □

Lemma 7.28 and Proposition 7.34 imply that ψ sends the stabilizers of blue vertices in T to the stabilizers of blue vertices in S. Consider components C and B of Õ − Σ̃_{O} and Q̃ − Σ̃_{Q} such that

ψ : Stab(C) → Stab(B)

The quotients cl(C)/Stab(C) and cl(B)/Stab(B) are naturally homeomorphic to |O|, |Q| respectively. According to Corollary 7.33, an element g ∈ Stab(C) stabilizes a boundary component of C iff ψ(g) stabilizes a boundary component of B. Thus the induced isomorphism |ψ| : π_{1}(|O|) → π_{1}(|Q|) carries peripheral elements to peripheral elements. Our next problem is to construct a natural homeomorphism ∂|O| → ∂|Q| which preserves the orbifold structures and is well-behaved with respect to the homomorphisms π_{1}(∂|O|) → π_{1}(|O|), π_{1}(∂|Q|) → π_{1}(|Q|).

As we already know each face F ⊂ F ix(r) of the component C ⊂ Õ − Σ̃_{O} is simplyconnected and there exists a homeomorphism h_{F} : F → F′ ⊂ F ix(ψ(r)) which conjugates the stabilizer of F to the stabilizer of F′.

Recall also that faces F_{1}, F_{2} are adjacent iff the corresponding reflections satisfy the relation (r_{1}r_{2})^{2} = 1. For adjacent faces (after adjusting homeomorphisms h_{F_{j}} via ψ-equivariant isotopy) we have:

h_{F_{1}}|_{F_{1}∩F_{2}} = h_{F_{2}}|_{F_{1}∩F_{2}}

Thus we define the homeomorphism h : ∂ÕC → ∂Q̃B as the union of homeomorphisms h_{F_{j}}. It is clearly well-defined and extends naturally to squares and planes which are lifts of components of ∂O, ∂Q. The resulting homeomorphism h is ψ-equivariant and it projects to a homeomorphism ĥ : ∂|O| → ∂|Q|. By construction, the induced isomorphisms of fundamental groups of boundary components of ∂|O| and ∂|Q| preserve the peripheral structure in the sense of Section 2.4. Thus by Theorem 2.33 there exists an extension of ĥ to a homeomorphism f : |O| → |Q| which preserves the orbifold structure on the boundary because ĥ did. □

8 Complex Projective Structures

In this section we consider complex projective structures on Riemann surfaces. These structures provide a generalization of Kleinian groups and they will be useful in our discussion of the geometric convergence in Section 9.14. I will give here only a general introduction to the theory of complex projective structures, for details and recent developements see [88], [104], [107], [72], [77], [191], [190].

8.1 Basic definitions

Let S be a smooth surface without boundary. A complex projective structure σ on S is a maximal atlas such that all transition maps belong to the group P SL(2, ℂ). Each complex projective structure on S defines a local diffeomorphism dev from the universal covering S̃ to the extended complex plane \overline{ℂ}. Locally the map dev is a complex–projective diffeomorphism with respect to the complex projective structures on S̃ and \overline{ℂ}. The map dev is called a developing map of σ. Assume that the fundamental group F = π_{1}(S) acts on S̃ as the group of covering transformations. Then the there is a homomorphism ρ : F → P SL(2, ℂ) so that dev is ρ-equivariant. The representation ρ is called a monodromy representation of the structure σ. This representation is unique up to conjugation in P SL(2, ℂ). The group ρ(F) is called the monodromy group. Conversely, if we are given a local homeomorphism dev : S̃ → \overline{ℂ} which is ρ-equivariant, then we can take the pull back dev^{∗}(can) of the canonical complex projective structure from \overline{ℂ} to S̃. The group F acts as a group of automorphisms of dev^{∗}(can), thus the projection of dev^{∗}(can) to S̃/F is a complex projective structure σ. The map dev is a developing map of this structure. An important class of complex projective structures is given by uniformization. Suppose that Γ ⊂ P SL(2, ℂ) is a Kleinian group acting discontinuously and freely on a domain D ⊂ \overline{ℂ}. Then the canonical complex projective structure on D projects to a complex projective structure on S = D/Γ which is uniformized by Γ. In this case the monodromy representation is an epimorphism ρ : π_{1}(S) → Γ with the kernel π_{1}(D). The developing map is the covering S̃ → D. However complex projective structure does not have to appear this way. In particular, the monodromy representation can be nondiscrete and image of the developing map can be the whole sphere \overline{ℂ}. Similarly to the definition of Teichmüller space one can define the space MCP (S) of all marked complex projective structures on S. Namely, we fix a complex projective structure s on S, then points in MCP (S) are equivalence classes of marked complex projective surfaces, which are triples (Σ, σ, f), where (Σ, σ) is a surface homeomorphic to S, σ is a complex projective structure on Σ and f : (S, s) → (Σ, σ) is an orientation-preserving homeomorphism. We identify two triples (Σ, σ, f), (Σ′, σ′, f′) if there is a complex projective diffeomorphism ϕ : Σ → Σ′ such that ϕ ∘ f is homotopic to f′.

Proposition 8.1 Suppose that two triples (Σ, σ, f), (Σ′, σ′, f′) are equivalent via a mapping ϕ : Σ → Σ′. Let d and d′ be developing maps for these marked surfaces. Then there exists a Moebius transformation ξ ∈ P SL(2, ℂ) such that ξ ∘ d′ ∘ ϕ = d.

Proof: Follows directly from the definition of the developing map. □

There is an alternative point of view on complex projective structure coming from the complex analysis. Let S be a smooth compact oriented surface of genus g ≥ 2. Each marked complex projective structure σ on S (which is consistent with the orientation on S) also determines a marked complex structure p(σ) ∈ T (S). Thus we get a natural projection MCP (S) → T (S) which is easily seen to be a continuous map. Now fix τ = p(σ) and consider the fiber p^{−1}(τ) ⊂ MCP (S). This fiber can be described as follows. Uniformize τ by a Fuchsian group F, (S, τ) = ℍ^{2}/F. Then for each ν ∈ p^{−1}(τ) the developing map f = dev_{ν} of ν is a locally injective holomorphic mapping ℍ^{2} → \overline{ℂ} which is ρ-equivariant where ρ is a monodromy homomorphism of ν. Then the Schwarzian derivative

S(f) = (\frac{f′′}{f′})′ − \frac{1}{2}(\frac{f′′}{f′})^{2} = \frac{f′′′}{f′} − \frac{3}{2}(\frac{f′′}{f′})^{2}

determines a holomorphic quadratic differential φ̃dz^{2} on ℍ^{2} which is F -invariant. Hence φ̃dz^{2} projects to a holomorphic quadratic differential φdz^{2} on the surface (S, τ). Viceversa, if φdz^{2} is a holomorphic quadratic differential on the surface (S, τ) then we lift it to an F -invariant holomorphic quadratic differential φ̃dz^{2} on ℍ^{2}. Then solve the Schwarzian differential equation

S(f) = φ̃

Each solution if locally injective holomorphic map f : ℍ^{2} → \overline{ℂ}, which is unique up to post-composition with γ ∈ Isom(ℍ^{3}) and which is therefore ρ-equivariant with respect to a homomorphism ρ : F → Isom(ℍ^{3}). This defines a map MCP (S) → Q(S) where Q(S) is the affine bundle over T (S) whose fiber over τ ∈ T (S) is the space of holomorphic quadratic differentials on (S, τ). One can easily show that this map is a homeomorphism. (It is actually a real-analytic diffeomorphism.) Thus we can identify MCP (S) with Q(S) ≅ T (S) × Q_{0}, where Q_{0} is the vector space of holomorphic quadratic differentials on (S, τ_{0}) for a fixed conformal structure τ_{0} on S. Therefore topologically MCP (S) is just ℂ^{6g−6}.

8.2 The holonomy theorem

Here is another way to think about complex projective structures . Given a representation ρ we construct the associated bundle E = S ×_{ρ} \overline{ℂ} as the quotient of S̃ × \overline{ℂ} by the following action of F:

γ(q, z) = (γ(q), ρ(γ)(z)), q ∈ S̃, z ∈ \overline{ℂ}, γ ∈ F

The bundle E = E_{ρ} has a natural flat connection ω = ω_{ρ}. Smooth maps S̃ → \overline{ℂ} which are ρ-equivariant, naturally correspond to sections of the bundle E. Such a map is a local diffeomorphism iff the corresponding section is transversal to ω.

Properties of monodromy representations. Suppose that S is a closed orientable surface of genus ≥ 2, σ is a complex projective structure on S with the monodromy representation ρ : π_{1}(S) → P SL(2, ℂ). Then:

1. ρ lifts to a representation ρ̃: π_{1}(S) → SL(2, ℂ).

2. ρ is always a nonelementary representation.

3. The bundle E_{ρ} is always trivial.

It turns out that the first two conditions are sufficient for ρ being the monodromy of a complex projective structure on S (see [107], [72]).

Therefore, instead of thinking about complex projective structures , one can work with the pair (ω, d), where ω is a flat connection on E = S×\overline{ℂ} and d is a smooth section of E which is transversal to ω, this section will be called the developing section (see Theorem 6.2). This allows us to topologize the space MCP (S) of marked complex projective structures on S by using C^{1} topology on the space of section of E and C^{1} topology on the space of connections on E.

There is a natural map hol : MCP (S) → R(F, P SL(2, ℂ)) which associates with each complex projective structure the conjugacy class of its monodromy representation. The following result is crucial for applications of complex projective structures in the theory of Kleinian groups.

Theorem 8.2 The map hol : MCP (S) → R(F, P SL(2, ℂ)) is a local homeomorphism.

Proof: This theorem was first proven by D. Hejhal [88] (see [55], [76] for generalizations). We shall prove here only that hol is an open map, our proof follows [76]. Suppose that [ρ] = hol(σ), let ω denote the corresponding flat connection and d the section of the bundle E. Let [ρ_{n}] ∈ R(F) be a sequence convergent to [ρ]. Then we have C^{1}–convergence of the corresponding flat connections ω_{n} → ω. Choose d_{n} = d to be the developing section. Therefore for large values of n the section d is transversal to ω_{n} and we take σ_{n} = [ω_{n}, d_{n}] ∈ MCP (S). □

Remark 8.3 Similar theorem holds for complex projective structures on Riemann surfaces of finite type. To make it work consider only those complex projective structures on S which near punctures look like cusps of complete hyperbolic surfaces. Instead of the representation variety R(F, P SL(2, ℂ)) one takes the relative representation variety R(F, P; P SL(2, ℂ)) where P is the collection of peripheral subgroups in F.

8.3 Grafting

In this section we discuss a very basic example of complex projective structure which has surjactive developing map and discrete monodromy. The first example of this kind was constructed by B.Maskit [132]. We start with a hyperbolic surface S and a simple closed geodesic γ ⊂ S. Let σ denote the hyperbolic complex projective structure on S. For sufficiently small ϵ > 0 the ϵ-neighborhood Nbd_{ϵ}(γ) of γ in S diffeomorphic to the annulus. We shall assume that the universal cover of S is embedded in \overline{ℂ} as the hyperbolic upper half-plane ℍ^{2}. Let g ∈ F = π_{1}(S) be the element of the fundamental group corresponding to the generator of π_{1}(γ). The quotient Ω(⟨g⟩)/⟨g⟩ is the torus T^{2} which has complex projective structure τ uniformized by the group ⟨g⟩, this torus contains a copy γ′ of the closed loop γ. Cut the surface S and the torus T^{2} open along the geodesics γ and γ′. Now glue these surfaces together along their boundary loops using the identification γ ∼ γ′ so that the orientations of S and T^{2} agree. The resulting surface S′ is diffeomorphic to S and has a new complex projective structure gr_{σ}(γ) which restricts to the structures σ and τ on S and T^{2}. The structure gr_{σ}(γ) is said to be obtained from the hyperbolic structure on S via grafting along γ. If we identify the fundamental groups of S and S′ with the Fuchsian group F then the monodromy representation of gr_{σ}(γ) is the identity map F → F ↪ Isom(ℍ^{3}). The image of the developing map dev_{gr_{σ}(γ)} of the structure gr_{σ}(γ) intersects the limit set of F ⊂ P SL(2, ℝ) in a nonempty subset (which contains Ω(⟨g⟩) ∩ Λ(F). Therefore (by equivariance) dev_{gr_{σ}(γ)} is onto \overline{ℂ}. Instead of the structure τ on the torus T^{2} we could have taken its n-fold (n ≥ 1) cover τ(n) which is induced by an n-fold covering over Ω(⟨g⟩). Then we again glue S′ and the complex-projective cylinder (T^{2} − γ′, τ(n)). We get a complex projective structure gr_{σ}(nγ) on the surface S. More generally, if γ_{1}, ..., γ_{r} is a collection of simple closed disjoint geodesics on S and n_{1}, ..., n_{r} is a collection of positive integers we define the grafting gr_{σ}(n_{1}γ_{1}∪, ..., ∪n_{r}γ_{r}). The structure gr_{σ}(n_{1}γ_{1}∪, ..., ∪n_{r}γ_{r}) still has the identity monodromy representation and the surjective developing map S̃ → \overline{ℂ}.

Definition 8.4 Let G be a quasifuchsian subgroup of P SL(2, ℂ) without parabolic elements. Then Ω(G) = Ω_{1} ∪ Ω_{2} is the disjoint union of two topological disks. Then the projection of the canonical complex projective structure from \overline{ℂ} to Ω_{j}/G is called a quasifuchsian complex projective structure . Clearly we have two marked quasifuchsian structures with the monodromy G: Ω_{1}/G, Ω_{2}/G, these structures have opposite orientations as the marked complex projective structure . If S is a closed orientable hyperbolic surface then a representation ρ : π_{1}(S) → Isom(ℍ^{3}) is called quasifuchsian if it is faithful and its image is a quasifuchsian subgroup.

Similarly one defines grafting for quasifuchsian structures. (The definition actually can be made much more general.)

Theorem 8.5 Suppose that μ is a complex projective structure on S with the monodromy representation id : F ≅ F and the same orientation as the original hyperbolic structure σ on S. Then either μ = σ or μ is equivalent to a structure which is obtained from σ by grafting.

Proof: This theorem was proven by W. Goldman in [77]. □

The above theorem allows us to describe the topology of the space of complex projective structure on S with quasifuchsian monodromy as follows. Fix an orientation o and hyperbolic structure on S and let QF (S) denote the space of marked quasifuchsian cps s on S which are compatible with the orientation o. Let C denote the set whose elements are formal linear combinations

n_{1}γ_{1} + … n_{m}γ_{m}

where n_{j} are positive integers, γ_{1}, ..., γ_{m} are disjoint simple closed geodesics on S (0 ≤ m ≤ 3g − 3 where g is the genus of S). Give C the discrete topology. Then hol^{−1}QF (S) is naturally homeomorphic to to the direct product T (S) × C, where T (S) is the Teichmuller space of S. Namely, each structure μ ∈ hol^{−1}QF (S) equals gr_{σ}(n_{1}γ_{1} ∪ ...n_{m}γ_{m}) where σ is a quasifuchsian structure on S with the monodromy ρ and the orientation consistent with o. Then we map μ to (n_{1}γ_{1} + … n_{m}γ_{m}, σ). One can easily see that this map is a homeomorphism.

At the first glance the quasifuchsian structure and its grafting seem to be billions of light years apart. It turns out however that this is not quite the case:

Theorem 8.6 (C. McMullen, see [142].) Let γ be a simple closed loop. Then the closures of QF (S) and gr_{γ}(QF (S)) in MCP (S) have nonempty intersection.

9 Sociology of Kleinian Groups

9.1 Algebraic convergence of representations

Let G be a Lie group and Γ a finitely generated group. A representation ρ : Γ → G is said to be discrete if its image is a discrete subgroup of G. A representation is said to be faithful if it is a monomorphism.

Definition 9.1 Suppose that ρ_{j} is a sequence of representations of Γ to G. Then the sequence ρ_{j} is said to be algebraically convergent to a representation ρ iff for each g ∈ Γ we have:

lim_{n→∞} ρ_{j}(g) = ρ(g)  (26)

where convergence is understood in the topology of the Lie group G.

The topology of algebraic convergence is consistent with the usual topology of the representation variety R(Γ, Isom(ℍ^{n}))^{0}. This is the most intuitively obvious concept of convergence of representations. As we shall see, there are some other definitions of convergence which differ from this one.

Suppose that ρ : Γ → Isom(ℍ^{n}) is a discrete and faithful representation of a discrete group Γ ⊂ Isom(ℍ^{n}), then ρ is nonelementary iff the group Γ is nonelementary itself. More generally, let G be a Lie group, then representation ρ : Γ → G is said to be non-radical if ρ(Γ) does not contain nontrivial normal nilpotent subgroup. Clearly, if G = Isom(ℍ^{n}) then ρ is nonelementary if and only if it is non-radical. Let D(Γ, G) denote the space of G-conjugacy classes of discrete and faithful representations of Γ to the Lie group G (most of the time we will have G = Isom(ℍ^{n})). Again there is a problem with taking quotient, however since we will mainly consider nonelementary representations to Isom(ℍ^{n}), it does not matter.

Example 9.2 Let Γ_{n} = ⟨a, b_{n}⟩ be an Abelian subgroup of Isom(ℂ), where a : z ↦ z + 1, b_{n} : z ↦ z + p + i/n where n ∈ ℤ, p ∈ ℝ. Define the sequence of representations ρ_{n} : Γ_{1} → Γ_{n} so that a ↦ a, b_{1} ↦ b_{n}. Then the sequence {ρ_{n}} consists of discrete and faithful representations. There exists an algebraic limit ρ so that ρ(a) = a, ρ(b_{1}) : z ↦ z + p. On the other hand, this limit is always either nondiscrete or nonfaithful. Our current goal is to prove that this is an exceptional situation.

Lemma 9.3 Suppose that G is any semi-simple Lie group. Then arbitrarily small neighborhood U of {1} in G contains elements a, b which generate a free nonabelian group in G.

Proof: According to the classification of Lie groups, G must contain a Lie subgroup H which is locally isomorphic either to SL(2, ℝ) or to SO(3). Each of the groups SL(2, ℝ), SU(2) contains a free nonabelian subgroup generated by elements α, β. In the case of SL(2, ℝ) we can take this subgroup to be a Schottky subgroup. In the case of SO(3) we can take the “Hausdorff example” (see [44]):

α = ghgh^{2}, β = gh^{2}gh

g = ( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & −1 \end{matrix} ) , h = ( \begin{matrix}1 & 0 & 0 \\ 0 & −1/2 & \sqrt{3}/2 \\ 0 & −\sqrt{3}/2 & −1/2 \end{matrix} )

Now, we connect α and β by real–analytic curves α_{t}, β_{t} with the identity. The set of values of t when α_{t}, β_{t} generate a nonfree group is a countable subset in [0, 1]. Therefore we can choose α_{t}, β_{t} to be arbitrarily close to the identity. It follows that corresponding elements in H will also generate a free nonabelian subgroup. □

Theorem 9.4 Suppose that Γ is a finitely generated group, ρ_{i} is a sequence of discrete, faithful and non-radical representations of Γ to G, which is algebraically convergent to ρ : Γ → G. Then ρ is again discrete and faithful. In other words, D(Γ, G) is closed in R(Γ, G).

Remark 9.5 This theorem is called “Chuckrow’s theorem”, despite the fact that V. Chuckrow [39] proved it only for Kleinian subgroups in P SL(2, ℂ) and her arguments cannot be generalized to higher dimensions. A better name would be probably “Wielenberg’s theorem”, who proved it for representations to SO(n, 1) see [207].

Proof: We identify Γ with ρ_{1}(Γ) ⊂ Isom(ℍ^{n}) (so that ρ_{1} = id). Note that for the fixed Lie group G there is an upper bound ν on the “length” of nilpotent subgroups of G, i.e. if the (ν + 1)-fold commutator [… [[g_{1}, g_{2}], g_{3}] …, g_{ν+1}] is non-trivial then g_{1}, ..., g_{ν+1} generate a non-nilpotent subgroup of G.

(i) First suppose that ρ is not faithful. Pick any finite collection of elements g_{1}, ..., g_{m} ∈ K. Then for all sufficiently large i their images ρ_{i}(g_{j}), 1 ≤ j ≤ m, belong to Zassenhaus neighborhood U_{Z} of the identity in G. Hence the subgroup generated by g_{1}, ..., g_{m} is nilpotent. Therefore K is exhausted by nilpotent subgroups which means that K is itself nilpotent, which contradicts the assumption that ρ_{1} is non-radical.

(ii) Now let us assume that the group ρ(Γ) is not discrete. Consider the closure H = \overline{ρΓ} of this group in G (with respect to the classical topology). The closure must be a Lie subgroup in Isom(ℍ^{n}). Let H^{0} be the identity component of H, it is a normal subgroup of H. If H is not nilpotent then we can find elements ρ(g_{1}), … ρ(g_{n+1}) ∈ H ∩ U_{Z} such that the (ν + 1)-fold commutator [… [[ρ(g)_{1}, ρ(g)_{2}], ρ(g)_{3}] …, ρ(g)_{ν+1}] is non-trivial. On the other hand, for sufficiently large indices i the elements ρ_{i}(g_{j}), 1 ≤ j ≤ n + 1, belong to U_{Z}. Therefore the subgroup generated by ρ_{i}(g_{j}), 1 ≤ j ≤ n + 1, has to be nilpotent. This is impossible. Thus H^{0} is a nilpotent Lie group. Since the representation ρ : γ → H is faithful it follows that ρ^{−1}(H^{0}) is a normal nilpotent subgroup of Γ which again contradicts the assumption that ρ_{1} is non-radical. □

Remark 9.6 For generalization of this result to the case of manifolds of negative curvature see [131].

9.2 Geometric convergence

The notion of algebraic convergence of representations is nice and simple, however there are situations when it does not reflect changes in geometry of the convergent sequences of groups. As before let G be a Lie group.

Definition 9.7 Suppose that Γ_{j} ⊂ G is a sequence of closed subgroups. Then the geometric limit Γ_{∞} := Lim_{j→∞}^{g}(Γ_{j}) of this sequence consists of limits of all sequences γ_{j_{i}} ∈ Γ_{j}. In other words, Γ_{j} is convergent to Γ_{∞} in the Chabauty topology.

The following proposition generalizes Theorem 9.4 (compare [15]). I am grateful to I. Belegradek who noticed an error in the original version of this proposition.

Proposition 9.8 Let Γ be a finitely generated group (with the generating set {γ_{1}, ..., γ_{m}}) and ρ_{j} : Γ → G = Isom(ℍ^{n}) be a sequence of non-radical discrete representations, let Γ_{j} = ρ_{j}(Γ). Then the geometric limit Γ_{∞} of the sequence Γ_{j} is a discrete subgroup of G.

Proof: We argue as in the proof of Theorem 9.4: assume that Γ_{8} is not discrete. The geometric limit H := Γ_{∞} is automatically a closed subgroup of G. Let H^{0} be the identity component of H. Suppose that H^{0} is not nilpotent. Then we can find elements g_{1}, ..., g_{ν+1} ∈ H^{0} ∩ U_{Z} such that the ν-fold commutator

[… [[g_{1}, g_{2}], g_{3}] …, g_{ν+1}]

is non-trivial where ν is the nilpotent length of G. By definition of the geometric convergence the elements g_{i} are approximated by g_{i,j} ∈ Γ_{j}, hence for large n we have: g_{i,j} ∈ U_{Z}, therefore g_{i,n}, i = 1, ..., ν + 1 generate a nilpotent group since Γ_{j} is discrete. Contradiction. Hence H^{0} is nilpotent. Now there are two distinct cases:

Case (a). The group H^{0} is a nontrivial semi-simple group, i.e. H^{0} − {1} consists only of loxodormic and elliptic elements. Then, after passing to a subgroup of index 2 if necessary, we can assume that H^{0} is central in Γ_{∞}. Let R_{j} := Γ_{j} ∩ U_{Z}, then the group N_{j} generated by R_{j} is nilpotent. The limit of R_{j} in the Chabauty topology equals cl(H ∩ U_{Z}), in particular the subgroups N_{j} are nontrivial for large j. If j is sufficiently large then for each element r_{j} ∈ R_{j} and each generator γ_{i} of the group Γ we have:

[r_{j}, ρ_{j}(γ_{i})] ∈ U_{Z} ∩ Γ_{j} = R_{j}

Therefore ρ_{j}(γ_{i})r_{j}ρ_{j}(γ_{i})^{−1} ∈ N_{j}, which implies that the subgroup N_{j} is normal in Γ_{j}. Therefore the representation ρ_{j} is not non-radical.

Case (b). The group H^{0} is not semi-simple, i.e. it contains parabolic elements. Let V ⊂ W ⊂ U_{Z} be neighborhoods of identity in G such that cl(W) ⊂ U_{Z} and for each γ_{i}, i = 1, ..., m, we have:

ρ_{∞}(γ_{i})V ρ_{∞}(γ_{i})^{−1} ⊂ W

Define R_{j} and N_{j} as above. Then the intersection Γ_{j} ∩ V contains parabolic elements for large j and for each r ∈ Γ_{j} ∩ V we have

ρ_{j}(γ_{i})rρ_{j}(γ_{i})^{−1} ∈ U_{Z}

Thus ρ_{j}(γ_{i})N_{j}ρ_{j}(γ_{i})^{−1}∩N_{j} contains parabolic elements. Each parabolic element has unique fixed point at infinity, thus the maximal nilpotent subgroup of Γ_{j} containing N_{j} is normal in Γ_{j}. This implies that ρ_{j} is not non-radical. □

Remark 9.9 The same arguments work in the case of all rank 1 Lie groups G. I do not know what happens for semi-simple Lie groups of higher rank.

Theorem 9.10 ([16], [55]) The following properties are equivalent:

1. A sequence of discrete groups G_{n} ⊂ Isom(ℍ^{3}) is convergent geometrically to a group G;

2. The pointed manifolds (M(G_{n}), x_{n}) are convergent to (M(G), x) (in the quasi-isome-tric topology) for some choice of base-points.

Suppose that ρ_{n} : Γ → Γ_{n} ⊂ Isom(ℍ^{3}) is a sequence of discrete and faithful representations, which is algebraically convergent to a discrete and faithful representation ρ_{∞} : Γ → Γ_{∞}. There are three essentially equivalent ways to state the basic problem which can appear.

1. The quotient manifolds M(Γ_{n}) = ℍ^{3}/Γ_{n} do not necessarily converge in the quasiisometric topology to the quotient manifold M(Γ_{∞}) for any choice of base-points. (See Section 4.9 for the definition of quasi-isometric convergence.)

2. Hausdorff limit of the sequence of limit sets Λ(Γ_{n}) can be much bigger than the limit set of the group Γ_{∞}.

3. The geometric limit Lim_{n→∞}^{g}(Γ_{n}) could be strictly larger than the group Γ_{∞}.

This phenomenon was first discovered by Troel Jorgensen and I describe below the simplest example of it following [192]. Start with a sequence of loxodromic representations of a cyclic group which is convergent to a discrete parabolic group. Let w_{n} = n^{−2} + πi/n. Define the sequence of representations ϕ_{n} : ℤ → SL(2, ℂ) as:

a_{n} = ϕ_{n}(1) = ( \begin{matrix}exp(w_{n}) & n sinh(w_{n}) \\ 0 & exp(−w_{n}) \end{matrix} )

Therefore a = ϕ(1) = lim_{n→∞} ϕ_{n}(1) is the matrix:

ϕ(1) = ( \begin{matrix}1 & πi \\ 0 & 1 \end{matrix} )

However the limit of the sequence ϕ_{n}(n) is the matrix

b = ( \begin{matrix}−1 & −1 \\ 0 & −1 \end{matrix} )

Thus the geometric limit contains ℤ^{2}. One can see that in this example the geometric limit is exactly the group ℤ^{2}. However the limit sets of the groups ϕ(ℤ) and ℤ^{2} = Lim_{n→∞}^{g}ϕ(ℤ_{n}) coincide (they are equal to a single point). To construct a more sophisticated example one can use the “Klein Combination”. Namely, let c be a loxodromic transformation in \overline{ℂ} which which does not fix the point ∞. Then for sufficiently large power m the complements to the fundamental domains of ⟨d = c^{m}⟩ and ⟨a_{n}⟩ are disjoint for all m. A fundamental domain for the group ℤ^{2} = Lim_{n→∞}^{g}ϕ_{n}(ℤ) is a rectangle R of the size 1 × π. One can also assume that the complement to the fundamental domain Φ of ⟨d⟩ is contained in R. Then the groups ρ_{n}(ℤ ∗ ℤ) = ⟨a_{n}, d⟩ are Kleinian and ρ_{n} : ℤ ∗ ℤ → ⟨a_{n}, d⟩ are isomorphisms. The group H = ⟨a, b, d⟩ is isomorphic to (⟨a⟩ ⊕ ⟨b⟩) ∗ ⟨d⟩. Therefore it can not contain Γ = Z ∗ ℤ as a subgroup of finite index. On the other hand, H is contained in the geometric limit of the sequence ρ_{n}(ℤ ∗ ℤ). One can actually prove that the geometric limit is exactly the group H. The group Γ = ⟨a⟩ ∗ ⟨d⟩ is the algebraic limit of ρ_{n}(ℤ ∗ ℤ). This group is geometrically finite, therefore its limit set can not coincide with the limit set of H (which contains Γ).

Theorem 9.11 Suppose that Γ is a finitely generated group, ρ_{n} : Γ → Γ_{n} ⊂ Isom(ℍ^{3}) is a sequence of discrete and faithful representations which is convergent algebraically to ρ_{∞} : Γ → Γ_{∞}, where Γ_{∞} is Kleinian, nonelementary and Λ(Γ_{∞}) is not contained in a round circle. If the Hausdorff limit of limit sets of Γ_{n} is the limit set of Γ_{∞} then the geometric limit G of the groups Γ_{n} equals Γ_{∞}.

Proof: The group G is discrete, Λ(G) is contained in the Hausdorff limit of the limit sets Λ(Γ_{n}), hence Λ(G) ⊂ Λ(Γ_{∞}). On the other hand, G ⊃ Γ_{∞}. Thus Λ(Γ_{∞}) = Λ(G) and Theorem 6.89 implies that |G : Γ_{∞}| < ∞. We need to prove that the index equals to 1. Here is the reason:

Lemma 9.12 Suppose that g ∈ G is an element of infinite order such that ρ_{∞}(h) = g^{m} ∈ Γ_{∞} for some m ≠ 0. Then g ∈ Γ_{∞}.

Proof: To begin with, we can assume that h is a primitive element in Γ. Suppose that γ_{i} ∈ Γ_{i} is a sequence of elements converging to g. Note that d(γ_{i}^{m}, ρ_{i}(h)) → 0 in the Lie group Isom(ℍ^{3}). Thus Theorem 6.43 implies that for large i the elements γ_{i}^{m}, ρ_{i}(h) belong to the same elementary subgroup of Γ_{i}. Since ρ_{i} are discrete and faithful we conclude that g ∈ Γ_{∞}. □

This concludes the proof of Theorem 9.11 as well. □

T. Jorgensen and A. Marden [103] and K. Ohshika [160], proved the following converse to the Theorem 9.11:

Theorem 9.13 Suppose that the group Γ in Theorem 9.11 is the fundamental group of a complete hyperbolic surface of finite area, homomorphisms ρ_{n}, ρ_{∞} are type-preserving and the geometric limit G of the groups Γ_{n} equals the algebraic limit Γ_{∞}. Then limits sets of Γ_{n} are Hausdorff-convergent to the limit set of Γ_{∞}.

9.3 Isomorphisms of geometrically finite groups

Theorem 9.14 (D. Mostow [154], P. Tukia [201]). (a) Let ρ : G_{1} → G_{2} be an isomorphisms between two torsion-free geometrically finite groups in ℍ^{n}, which is type-preserving (i.e. g ∈ G_{1} is parabolic iff ρ(g) is parabolic). Then there is ρ-equivariant homeomorphism h : Λ(G_{1}) → Λ(G_{2}). (b) Moreover, if n ≥ 3 and X ⊂ ℍ^{n} ∪ Ω(G_{1}), f : X → ℍ^{n} ∪ Ω(G_{2}) is a quasiconformal embedding inducing ρ, then the extension of f via h is again quasiconformal.

We shall prove only the part (a) of the statement in the case when G_{1} has no parabolic elements, a proof of the whole theorem in the general case can be found in [201].

Proof: Suppose that Γ_{1}, Γ_{2} is a pair of groups as in Theorem 9.14. There exist two Lipschitz maps

f_{1} : C(M(Γ_{1})) → C(M(Γ_{2})) , f_{2} : C(M(Γ_{2})) → C(M(Γ_{1}))  (27)

which are homotopy-equivalences inducing the isomorphisms ρ : Γ_{1} → Γ_{2}, ρ^{−1} : Γ_{2} → Γ_{1}. Therefore, f_{1} ∘ f_{2} and f_{2} ∘ f_{1} are homotopic to identity. Thus there is a constant C such that the lifts f̃_{j} of f_{j} to the convex hulls C(Λ(Γ_{j})) satisfy the property:

d(f̃_{1} ∘ f̃_{2}(x), x) ≤ C  (28)
d(f̃_{2} ∘ f̃_{1}(y), y) ≤ C  (29)

for every x ∈ C(Λ(Γ_{2})), y ∈ C(Λ(Γ_{2})). Hence both maps f̃_{j} are quasi-isometries. The map f̃_{1} is ρ-equivariant. Therefore we can apply Theorem 4.28 to conclude that f̃_{1} admits a quasiconformal extension h : Λ(Γ_{1}) → Λ(Γ_{2}) which (by continuity) induces the isomorphism ρ. □

Theorem 9.15 Suppose that Γ ⊂ P SL(2, ℂ) is a finitely generated Kleinian group. Then the following 3 assertions are equivalent:

1. Γ is quasifuchsian;

2. there exists a Fuchsian group F ⊂ P SL(2, ℝ) and a quasiconformal homeomorphism f : \overline{ℂ} → \overline{ℂ} such that fΓf^{−1} = F;

3. Γ is geometrically finite and admits a type-preserving isomorphism ρ to a Fuchsian group F ⊂ PSL(2, ℝ).

Proof: The implication (2)⇒ (1) is obvious. Let’s prove (1)⇒ (3). Suppose that Γ is a quasifuchsian group, then it is topologically conjugate to a Fuchsian group F. The group F is geometrically finite and the homeomorphism f of limit sets preserves points of approximation and bounded parabolic points (see Section 3.7). Therefore each limit point of Γ is either a point of approximation or a bounded parabolic point. This implies that Γ is geometrically finite. Now, we prove (3) ⇒ (2).

By Theorem 9.14 there exists a homeomorphism h : Λ(F) → Λ(Γ) which induces ρ. Thus, the limit set of Γ is a topological circle and Ω(Γ) is the union of two open disks Ω_{1}, Ω_{2} . Each of these disks is invariant under Γ (since h preserves the orientation of the limit circles). Since Γ is finitely generated, the surfaces Ω_{j}/Γ have finite conformal type and there exists a quasiconformal homeomorphism α : Ω(F) → Ω(Γ) which induces the isomorphism ρ of the fundamental groups of connected components. The lift of α to a quasiconformal homeomorphism of domains of discontinuity

α̃ : Ω(F) → Ω(Γ)  (30)

extends via h to a quasiconformal homeomorphism of \overline{ℂ} (by Theorem 9.14) which conjugates F to Γ. Therefore the group Γ is quasi-Fuchsian. □

Let Γ ⊂ P SL(2, ℂ) be a geometrically finite group, S_{j} = Ω_{j}/Γ_{j}, where Γ_{j} is the stabilizer of a component Ω_{j} of Ω(Γ). We have a natural epimorphism θ_{j} : π_{1}(S_{j}) → Γ_{j}. An element γ ∈ Γ_{j} is called an accidental parabolic if θ^{−1}(γ_{j}) contains no elements corresponding to peripheral loops in S_{j}.

Corollary 9.16 Suppose that Γ ⊂ P SL(2, ℂ) is a geometrically finite Kleinian group so that stabilizers Γ_{j} of components of Ω(Γ) in Γ do not have accidental parabolic elements and S_{j} = Ω_{j}/Γ_{j} is incompressible in M̄(Γ). Then each Γ_{j} is a quasi-Fuchsian group.

Proof: Let Γ_{j} be a stabilizer of a component Ω_{j} of Ω(Γ). Then Γ_{j} is geometrically finite (by Corollary 6.78) and θ : π_{1}(S_{j}) → Γ_{j} is an isomorphism. However S_{j} = ℍ^{2}/F_{j}, where F_{j} is a Fuchsian group preserving the upper-half plane ℍ^{2}. Thus we can apply Theorem 9.15 to conclude that Γ_{j} is quasi-Fuchsian. □

Remark 9.17 It is easy to see that the condition ∂M̄(Γ) to be incompressible is necessary. For example, let Γ be a Schottky group. Then Γ satisfies all but one conditions of the Corollary, however the stabilizer of the only component of Ω(Γ) is Γ itself which is not quasi-Fuchsian (since its limit set is not a topological circle). Another class of examples appears when Γ_{j} have accidental parabolic elements (“regular B-groups”).

Theorem 9.18 (A. Marden, W. Thurston, H. Reimann). Suppose that ρ : G_{1} → G_{2} is an isomorphism of two subgroups in Isom(ℍ^{3}), f : \overline{ℂ} → \overline{ℂ} is a quasiconformal homeomorphism which is ρ-equivariant. (The groups do not have to be discrete). Then h admits a ρ– equivariant quasiconformal extension f̃ : \overline{ℍ^{3}} → \overline{ℍ^{3}}. Moreover K_{O}(f̃) ≤ K^{3}(f) and f̃ depends continuously on f (with respect to the topology of uniform convergence).

Proof: See [170]. □

Theorem 9.19 (A. Douady, C. Earle, [47]) Suppose that ρ : G_{1} → G_{2} is an isomorphism of two subgroups in Isom(ℍ^{2}) (which do not have to be discrete), h : 𝕊^{1} → 𝕊^{1} is a quasimoebius homeomorphism which is ρ-equivariant. Then f admits a ρ-equivariant quasiconformal extension into ℍ^{2}.

This theorem immediately implies Theorem 5.6. Indeed, according to Theorem 9.14, since the groups F, F′ are geometrically finite, there exists a ρ-equivariant quasimoebius homeomorphism h : 𝕊^{1} → 𝕊^{1}. By Theorem 9.19, h admits a ρ-equivariant quasiconformal extension to ℍ^{2}. □

9.4 Mostow Rigidity Theorem

Now we can prove the Mostow Rigidity Theorem 2.46. Suppose that Γ, Γ′ are isomorphic lattices in Isom(ℍ^{n}) and ρ : Γ → Γ′ is an isomorphism. Then ρ is type-preserving. (Indeed, parabolic elements of Γ, Γ′ are characterized by the property that their centralizers in Γ, Γ′ contain ℤ × ℤ.) Thus by Theorem 9.14 there exists a ρ-equivariant quasiconformal homeomorphism h : 𝕊^{n−1} → 𝕊^{n−1}. Quasiconformal homeomorphisms are differentiable a.e. on 𝕊^{n−1} and have a.e. non-zero Jacobian determinants. Suppose that z ∈ 𝕊^{n−1} is a point of differentiability so that J_{z}(h) ≠ 0. Since only countable number of points in 𝕊^{n−1} are fixed points of parabolic elements, we can choose z to be a point of approximation of Γ. The result will follow from the following:

Theorem 9.20 (P. Tukia, N. Ivanov) Suppose that Γ is a discrete nonelementary subgroup of Isom(ℍ^{n}), z ∈ Λ(Γ) is a point of approximation, h : 𝕊^{n−1} → 𝕊^{n−1} is a homeomorphism which conjugates Γ to another Moebius group Γ′ = hΓh^{−1}. If h is differentiable at the point z with nondegenerate derivative, then the map h is necessarily Moebius.

Proof: Our proof will follow arguments of [96] (see also [200], [176]), under the extra assumption that h is quasiconformal , which will suffice for our purposes. We consider the upper half-space model of ℍ^{n} so that 𝕊_{∞}^{n−1} ≅ \overline{ℝ^{n−1}}. By applying a conformal change of coordinates we can assume that z = h(z) = 0 ∈ ℝ^{n−1}. Let L ⊂ ℍ^{n} be the “vertical” geodesic emanating from 0, y_{0} ∈ L is a base-point. Since z is a point of approximation, there is a sequence of elements γ_{i} ∈ Γ so that γ_{i}(y_{0}) → z and d(γ_{i}(y_{0}), L) ≤ C for each i. Let y_{i} denote the nearest-point projection of γ_{i}(y_{0}) to L. Take the sequence of hyperbolic translations T_{i} : \overrightarrow{x} ↦ λ_{i}\overrightarrow{x} with the axis L, so that T_{i}(y_{0}) = y_{i}. Then the sequence γ_{i}^{−1}T_{i} is relatively compact in Isom(ℍ^{n}) and lies in a compact K_{0} ⊂ Isom(ℍ^{n}). Now we form the sequence

h_{i}(x) := λ_{i}^{−1}h(λ_{i}x) = T_{i}^{−1} ∘ h ∘ T_{i}(x)

Note that λ_{i} → 0 as i → ∞. Since the function h is assumed to be differentiable at zero, there is a linear transformation A ∈ GL(n − 1, ℝ) so that

lim_{i→∞} h_{i}(x) = Ax

for all x ∈ ℝ^{n−1}. Since the sequence h_{i} is K-quasiconformal , Property (4) of quasiconformal mappings in Section 5.1 implies that this convergence is uniform on compacts in ℝ^{n−1}. By assumption, h_{i} conjugates the group Γ_{i} := T_{i}^{−1}ΓT_{i} ⊂ Isom(ℍ^{n}) into the group of Moebius transformations. Since the sequence γ_{i}^{−1}T_{i} is relatively compact in Isom(ℍ^{n}), the groups Γ_{i}, γ_{i}^{−1}Γγ_{i} = Γ are uniformly Hausdorff close in Isom(ℍ^{n}), independently on i. Therefore the sequence Γ_{i} is convergent geometrically in Isom(ℍ^{n}) to a nonelementary group Γ_{∞}. Hence the limit A conjugates Γ_{∞} into Isom(ℍ^{n}). However the group Γ contains an element γ such that γ(∞) ≠ ∞.

Lemma 9.21 Suppose that γ ∈ Isom(ℍ^{n}) is such that γ(∞) ≠ ∞, A ∈ GL(n − 1, ℝ) is an element which conjugates γ to AγA^{−1} ∈ Isom(ℍ^{n}). Then A is a Euclidean similarity, i.e it belongs to SO(n − 1) · ℝ_{+}.

Proof: Let P be a hyperplane in ℝ^{n−1} which does not contain zero. Then γ ∘ A^{−1}(P) is a round sphere Σ in ℝ^{n−1}. A(Σ) is again a round sphere iff A is a similarity, otherwise AγA^{−1} does not send planes to round spheres and is not Moebius. □

So far, so good: we have proved that h is conformal at the point z. To conclude that h is Moebius we need to use the fact that hΓh^{−1} ⊂ Isom(ℍ^{n}) once again.

Pick 3 distinct points a, b, c ∈ 𝕊^{n−1}. A normalization N(f) of a homeomorphism f : 𝕊^{n−1} → 𝕊^{n−1} is a postcomposition g ∘ f, where g ∈ Isom(ℍ^{n}), so that N(h) fixes {a, b, c} pointwise. The normalization is uniquely determined by f up to postcomposition with an element of the compact subgroup K_{abc} ⊂ Isom(ℍ^{n}) which fixes the round circle in 𝕊^{n−1} containing {a, b, c}. We shall consider N(f) as an element of the coset K_{abc} · Homeo(𝕊^{n−1}). Thus, N(g ∘ f) = N(f) for all g ∈ Isom(ℍ^{n}). The fact that hΓh^{−1} ⊂ Isom(ℍ^{n}) means that N(h) ∘ γ ≡ N(h ∘ γ) = N(h) for all γ ∈ Γ. Recall that T_{i} = γ_{i} ∘ k_{i} where k_{i} are elements of a compact K_{0} ⊂ Isom(ℍ^{n}). Thus

N(h_{i}) = N(T_{i}^{−1} ∘ h ∘ T_{i}) = N(h ∘ T_{i}) = N(h ∘ γ_{i} ∘ k_{i}) = N(h) ∘ k_{i}

since γ_{i} ∈ Γ. On the other hand,

lim_{i→∞} N(h) ∘ k_{i} = lim_{i→∞} N(h_{i}) = N(A)

where A ∈ Isom(ℍ^{n}) is a Euclidean similarity. Hence there exists a relatively compact sequence k′_{i} ∈ K_{abc} so that

lim_{i→∞} k′_{i} ∘ N(h) ∘ k_{i} = N(A)

Extracting a subsequence and taking the limit we conclude that

k′ ∘ N(h) ∘ k = N(A)

which means that h ∈ Isom(ℍ^{n}). This finishes the proof of Theorem 9.20. □

The following is a simple but useful corollary of Theorems 2.46 and 2.34:

Theorem 9.22 Suppose that (M, P) is a paired Haken manifold so that one of its finite coverings (N, Q) admits a complete hyperbolic structure of finite volume. Then (M, P) also admits a complete hyperbolic structure of finite volume.

Proof: Without loss of generality we can assume that the covering N → M is regular, let F denote the group of covering transformations. Each of the elements f ∈ F is homotopic to an isometry f^{#} by Mostow Rigidity Theorem. Moreover f^{#} ∘ g^{#} = (f ∘ g)^{#} since any isometry of N which is homotopic to the identity is the identity itself. Thus we get an action of a finite group of isometries F^{#} on N which induces the same outer automorphisms of π_{1}(N) as F. Note that the action of F^{#} is free. Indeed, suppose that some f^{#} has a nonempty fixed-point set. Lift f and f^{#} to the universal cover ℍ^{3}. The lift is not unique, but we can compose lifts with elements of G = π_{1}(N) ↷ ℍ^{3} to guarantee that:

• The extensions of the lifts f̃^{#} and f̃ to the sphere at infinity \overline{ℂ} coincide.

• f̃^{#} is an elliptic isometry of finite order m.

Thus f̃^{m} ∈ G and acts trivially on \overline{ℂ}, whence f̃: ℍ^{3} → ℍ^{3} has finite order. We conclude that the group ⟨f̃⟩ does not act freely on ℍ^{3}, which contradicts the fact that F acts freely on N.

Consider the complete lifts of the groups F, F^{#} to ℍ^{3}: these lifts Γ, Γ^{#} are groups of diffeomorphisms of ℍ^{3}, and they are finite normal extensions of the group G. All elements of F, F^{#} induce nontrivial automorphisms of G. Extend both actions to \overline{ℂ}: restrictions of Γ, Γ^{#} to \overline{ℂ} are faithful representations Γ, Γ^{#} → Homeo(\overline{ℂ}) and they have the same image. We conclude that the groups Γ, Γ^{#} are isomorphic. Hence the manifolds M = N/F and M^{#} = N/F^{#} have isomorphic fundamental groups (≅ Γ) and they must be homeomorphic by Theorem 2.34. □

9.5 Sullivan Rigidity Theorem

Suppose that Γ ⊂ Isom(ℍ^{n}) is a discrete group whose action on its limit set is recurrent. Let Γ′ ⊂ Isom(ℍ^{n}) be another discrete group and f : ∂_{∞}ℍ^{n} → ∂_{∞}ℍ^{n} be a quasiconformal homeomorphism which conjugates Γ to Γ′.

Theorem 9.23 (Sullivan rigidity theorem). If under the conditions above the restriction of f to Ω(Γ) is conformal, then f is a Moebius transformation.

Proofs of this theorem can be found in [187], [5], [162].

We already know that finitely generated discrete subgroups of Isom(ℍ^{3}) act recurrently on their limit sets. Hence

Corollary 9.24 (D. Sullivan) Suppose that G ⊂ P SL(2, ℂ) is a finitely generated discrete group and μ is a G-invariant Beltrami differential such that supp(μ) ⊂ Λ(G). Then μ = 0 almost everywhere.

The following is one of the many characterizations of recurrent groups:

Theorem 9.25 (D. Sullivan) A discrete group Γ ⊂ Isom(ℍ^{n}) is recurrent if and only if for some (any) convex fundamental polyhedron P for the action of Γ in ℍ^{n}, the intersection

∂_{∞}P ∩ Λ(Γ)  (31)

has zero Lebesgue measure.

For proof see [187]. We shall apply this theorem in the situation when the intersection (31) is countable.

9.6 Bers isomorphism

In this section we consider a discrete, nonelementary, finitely generated groups G ⊂ Isom(ℍ^{3}) whose limit sets are connected. Equivalently: ∂M̄(G) is incompressible in M̄ (in the torsionfree case). We will consider only those quasiconformal homeomorphisms which preserve orientation. Define the Teichmüller space of the group G as

T (G) = {ρ ∈ Hom(G, Isom(ℍ^{3})) : there is a quasiconformal homeomor-

phism f : \overline{ℂ} → \overline{ℂ} such that f ∘ γ = ρ(γ) ∘ f for each γ ∈ G\}/P SL(2, ℂ)

If fGf^{−1} ⊂ P SL(2, ℂ) then the homomorphism f_{∗}(γ) = f ∘ γ ∘ f^{−1}, f_{∗} : G → P SL(2, ℂ) is called a representation induced by f.

For each G-conjugacy class of maximal parabolic subgroups in G choose a representative P_{j}. Let P = {P_{1}, P_{2}, ...} be the collection of representatives. I recall that the relative representation variety R(G, P; Isom(ℍ^{3})) consists of conjugacy classes of representations which send elements of P_{j} to parabolic transformations. Thus T (G) ⊂ D(G, Isom(ℍ^{3})) ∩ R(G, P; Isom(ℍ^{3})).

Let L_{∞}(Ω(G)) be the space of complex-valued L_{∞}-functions on the domain Ω(G). The condition that the element f_{∗}(γ) = f ∘γ ∘f^{−1} is a Moebius transformation can be expressed in terms of Beltrami differential μ(z) of the quasiconformal homeomorphism f:

μ(z) = μ(γz)\overline{γ′(z)}/γ′(z)  (32)

If μ satisfies the (32) for all γ ∈ G, then μ is G-invariant. Let

B_{∞}(G) := {μ ∈ L_{∞}(Ω(G)) : ||μ||_{∞} < 1, μ is G-invariant}

This space is the open unit ball in the Banach space

L_{∞}(G) := {μ ∈ L_{∞}(Ω(G)) : μ is G-invariant}

Thus we can think about T (G) as the quotient of the space B_{∞}(G) by the following equivalence relation:

μ ∼ ν iff the quasiconformal homeomorphisms f_{μ}, f_{ν} induce

the same element of R(G, Isom(ℍ^{3}))

We assume that T (G) has the topology of algebraic convergence induced from the usual topology of the algebraic variety R(G, Isom(ℍ^{3})). Let d_{T} denote the “Teichmüller distance” between two points [ρ], [ϕ] ∈ T (G), which is defined as

d_{T} (ρ, ϕ) = inf\{\frac{1}{2} log K(f) : f : \overline{ℂ} → \overline{ℂ} is a quasiconformal homeomorphism
such that f_{∗}(γ) = ϕ ∘ ρ^{−1}(γ) for each γ ∈ ρ(G)}

We denote by T_{t}(G) the space T (G) provided with the topology induced by the Teichmüller metric.

From now on let me assume that the group G is torsion-free: it will simplify our arguments and terminology.

Let Σ = Ω(G)/G = Σ_{1} ∪ ... ∪ Σ_{n} be the disjoint union of connected components. Define the Teichmüller space T (Σ) as the direct product of Teichmüller spaces T (Σ_{1})×...×T (Σ_{n}). We assume that the Teichmüller metric on the product is the maximum of Teichmüller metrics on components.

Theorem 9.26 There exists a natural homeomorphism β : T (Σ) → T_{t}(G) which is called the Bers identification. This homeomorphism is an isometry of Teichmüller metrics. The identity map T_{t}(G) → T (G) is always continuous. If G is geometrically finite then the identity map T_{t}(G) → T (G) is a homeomorphism.

Remark 9.27 If we drop the assumption that ∂M̄(G) is incompressible then the map β is a covering map.

Proof: A detailed proof of this result can be found in the paper of Bers [17]. Here I present only a sketch.

We can assume that the group G is Kleinian (otherwise the assertion follows from the Sullivan Rigidity Theorem). Denote by X the surface Σ as the origin of the Teichmüller space. Let [f : X → Y] ∈ T (Σ), where f is a quasiconformal map. Then the Beltrami coefficient ¯∂f/∂f of f is a Beltrami differential μ on X. Lift μ to a differential μ̃ on Ω(G) using the covering Ω(G) → X. Then μ̃ is invariant under G and we extend μ̃ by zero to the limit set of G. The result is a Beltrami differential on \overline{ℂ} which is G-invariant and we solve the Beltrami equation

¯∂h/∂h = μ  (33)

The quasiconformal homeomorphism h will conjugate G to a Moebius group G′. Therefore, the induced representation ρ = h_{∗} of the group G is an element of T (G). We put [ρ] = β[f : X → Y]. The right inverse map works as follows. Suppose that [ρ] ∈ T (G), then we let G′ = ρ(G), Y = Ω(G′)/G′ and the marking X → Y is obtained by descending a ρ-equivariant mapping f from Ω(G) → Ω(G′) to X → Y.

Let’s check that this correspondence is correctly defined. Suppose that [f : X → Y] = [g : X → Z] is a point in the Teichmüller space of X. Then there exists a biholomorphic map α : Z → Y which is homotopic to f ∘ g^{−1}. Define the representations ρ′ and ρ′′ using β(f) and β(g). The corresponding groups G′ and G′′ are finitely generated and have the property that the representation ρ′ ∘ (ρ′′)^{−1} is induced by a quasiconformal map α̃ which is conformal on Ω(G′′). By Sullivan’s rigidity theorem, the map α̃ is Moebius. Thus [ρ′] = [ρ′′]. The fact that β is onto follows from existence of a right inverse to [β]. Then we should check that β is injective. Suppose that β[X, id] = β[f : X → Y]. The lift of f to Ω(G) defines a quasiconformal homeomorphism h : \overline{ℂ} → \overline{ℂ} which conjugates G to itself and induces a trivial automorphism of G. Since all components of Ω(G) are simply-connected this implies that the projection of h to X is homotopic to a conformal map. Sullivan’s rigidity theorem implies that β is an isometry of Teichmüller metrics on T (G) and T (X). This proves the first part of the theorem.

The fact that id : T_{t}(G) → T (G) is continuous follows from continuity of solutions of the Beltrami equation with respect to the Beltrami differentials.

The bicontinuity of this map in the case of geometrically finite groups is less obvious. One way to prove it is to establish that the relative representation variety R(G, P; Isom(ℍ^{3})) is smooth near any point [ρ] ∈ T (G) and has the same dimension as T (X). (We shall give a sketch of the proof below in Sections 9.7, 9.8.) Then continuity of β^{−1} follows from the Browder’s theorem on invariance of domain. Another way is to establish so called “quasiconformal stability of geometrically finite groups” using direct geometric arguments (see [129], [189]). □

Remark 9.28 If Σ is orientable, then the Teichmüller spaces

T (G) ⊂ R(G, P; Isom(ℍ^{3})) and T (Σ)

have natural complex structures (for the complex structure on T (Σ) see my notes from the last year). One can prove that β is biholomorphic.

9.7 Smoothness of the representation variety

To simplify the proof let me assume here that the group G ⊂ P SL(2, ℂ) has no parabolic elements. Then the surface Σ is compact. The (complex) dimension of T (Σ) is equal to 3|χ(Σ)|/2. Consider a point [ρ : G → Γ] ∈ D(G, P SL(2, ℂ)), we do not assume that Γ is a geometrically finite group, however I will assume that Γ has no parabolic elements.

As far as I know, no proof of the following result is available in the existing literature unless the group is question is supposed to be geometrically finite.

Theorem 9.29 Suppose that Γ is a finitely generated discrete torsion-free subgroup of P SL(2, ℂ) as above. Then the identity representation id : Γ ↪ Isom(ℍ^{3}) determines a smooth point of the representation variety R(Γ, Isom(ℍ^{3}))

Proof: Let G denote the Lie algebra of the group Isom(ℍ^{3}); we shall consider G as a Γmodulus via the adjoint action of Γ. Suppose that Γ splits into free product of freely indecomposable factors Γ_{j}, j = 1, ..., s. Then Hom(Γ, Isom(ℍ^{3})) = Hom(Γ_{1}, Isom(ℍ^{3})) × ... × Hom(Γ_{s}, Isom(ℍ^{3})). So, its enough to consider only freely indecomposable groups. Choose a Scott compact core N := M_{c} in the manifold M = ℍ^{3}/Γ. N has incompressible boundary and is a deformation retract of M. Our goal is to prove that the second group cohomology H^{2}(Γ, G) is equal to zero. It is well– known that H^{2}(Γ, G) ≅ H^{2}(N, adP), where adP is the sheaf of parallel sections of the flat G–bundle over N associated with the adjoint action of Γ on the Lie algebra G. Another interpretation of the groups H^{j}(N, adP) is that they are cohomologies of the de Rham complex of differential forms on N with values in the bundle adP (see [169], [78]). By the Poincare duality we conclude that H^{2}(N, adP) ≅ H^{1}(M, M \ N; adP). However, each component C_{j} of ∂N is a closed hyperbolic surface (since Γ has no parabolic elements) and π_{1}(C_{j}) operates discretely and effectively on ℍ^{3}. Therefore, for each component C_{j} ⊂ M − N we have: H^{0}(C_{j}, adP) = 0, since π_{1}(C_{j}) is a nonelementary subgroup of P SL(2, ℂ). The long exact sequence of the pair (M, N) and the Poincare duality imply that

H^{1}(M, M \ N; adP) ≅ Ker[H^{1}(M, adP) → H^{1}(M \ N; adP)]  (34)

Suppose [ξ] is an element of the kernel. Then we can take ξ to be a differential 1-form on M with values in adP so that ξ is exact in each component of M \ N. This means that there is a smooth section η : M \ N → adP so that ζ = ξ − dη is zero in M \ N. Using a partition of unity we extend η to a smooth section η : M → adP. Hence the 1-form ζ = ξ − dη is cohomologous to ξ in M and vanishes identically in M − N. The form ζ has compact support in int(N), which implies that ζ belongs to L^{2}(M, adP); therefore it is exact by [169], Proposition 7.57. Hence

Ker[H^{1}(M, adP) → H^{1}(M \ N; adP)] = 0

This concludes the proof of the fact that H^{2}(Γ, G) = 0. Thus, by [125] or [78], we have proved that the variety R(Γ, Isom(ℍ^{3})) is smooth at the point [id]. □

Remark 9.30 With some extra care one can generalize the above arguments to the case of groups with parabolic elements.

9.8 Calculation of dimension of the representation variety

Now we can calculate complex dimension of the variety R(Γ, P SL(2, ℂ)) near [id]. First of all it must be equal to the dimension of H^{1}(N, adP) where N is the compact core as above. We know that H^{3}(N, adP) = H^{2}(N, adP) = H^{0}(N, adP) = 0 since ∂N ≠ ∅. Thus complex dimension dim_{ℂ}H^{1}(Γ, G) = χ(Γ, G) which is the Euler characteristic of Γ with twisted coefficients. The last is equal to 3χ(N), where χ(N) is the usual Euler characteristic since 3 = dim_{ℂ}SL(2, ℂ). However, χ(N) = χ(∂N)/2 since the Euler characteristic of the double of N along the boundary is zero. We conclude that dim_{ℂ}R(Γ, Isom(ℍ^{3})) near [id] is equal to 3χ(∂N)/2. Consider the case when Γ ∈ T (G) is a geometrically finite group without parabolic elements and Λ(Γ) is connected. As the manifold N we can take an ϵneighborhood of the Nielsen’s convex core C(Λ(Γ))/Γ. The boundary of N is homeomorphic to Σ = Ω(Γ)/Γ. Therefore, we conclude that dim_{ℂ}R(Γ, Isom(ℍ^{3})) = 3χ(Σ)/2. This means that complex dimension of the variety R(Γ, Isom(ℍ^{3})) near any representation [ρ] ∈ T (Γ) is equal to the dimension of T (G) and R(Γ, Isom(ℍ^{3})) is smooth at such points. Bers’ theorem follows.

Theorem 9.31 Suppose that G ⊂ Isom(ℍ^{3}) is a torsion–free geometrically finite group without parabolic elements. Then the Teichmüller space T (G) is open in R(G, Isom(ℍ^{3})) (with respect to the topology of algebraic convergence).

Proof: Follows from the preceding calculations since both are manifolds and have the same dimension. □

Corollary 9.32 Suppose that G is a finitely generated group,

[ρ_{n}] ∈ R(G, Isom(ℍ^{3}))

is a sequence of representations of G which is convergent to a faithful representation ρ : G → Γ ⊂ Isom(ℍ^{3}) such that Γ is a torsion–free geometrically finite group without parabolic elements. Then for all sufficiently large numbers n, the representations ρ_{n}∘ρ^{−1} : Γ → ρ_{n}(G) are induced by quasiconformal conjugations f_{n} : \overline{ℂ} → \overline{ℂ} and

lim_{n→∞} K(f_{n}) = 1

Proof: This is just another reformulation of Theorems 9.31 and 9.29.

The property of the group Γ above is called the quasiconformal stability. Sullivan proved that this property is satisfied (in the class of finitely-generated torsion-free groups) if and only if the group G is convex-cocompact [189]. Below we present a proof based on Theorem 9.7 and Sullivan’s rigidity theorem.

Theorem 9.33 Suppose that G is a finitely-generated discrete torsion-free subgroup of P SL(2, ℂ) which contains no parabolic elements. Then G is quasiconformally stable iff G is convex-cocompact.

Proof: We already know that any convex-cocompact group is quasiconformally stable. Suppose that G is a quasiconformally stable group. The space of quasiconformal deformations T (G) is isomorphic to the Teichmüller space T (Ω(G)/G). By the assumption this space is locally homeomorphic to the representation variety R(G, Isom(ℍ^{3})) near the identity representation. In particular they must have the same dimensions. Take a Scott compact core N of the manifold M = ℍ^{3}/G. If M has a geometrically infinite end, then |χ(∂N)| is strictly bigger than |χ(Ω(G)/G)|. However, according to our calculation of dimensions

\frac{3}{2}|χ(∂N)| = dim_{ℂ}R(G, Isom(ℍ^{3})) = dim_{ℂ}T (Ω(G)/G) = \frac{3}{2}|χ(Ω(G)/G)|

Thus |χ(∂N)| = |χ(Ω(G)/G)| and hence M has no geometrically infinite ends. This proves that G is convex-cocompact. □

9.9 Space of quasi-fuchsian representations

The most important example for us will be the case when G is a Fuchsian group. Then Ω(G) = Ω∪Ω̄ where the overline does not means “the closure” but the complementary open topological disk in 𝕊^{2} (this notation will become convenient later). Thus Σ = Ω(G)/G = Σ ∪ Σ̄ = Ω/G ∪ Ω̄/G. We can consider Σ, Σ̄ as two copies of one and the same Riemann surface (of genus g) with different orientations. Then T (G) = T (Σ)×T (Σ̄) and has complex dimension 6g−6+2p, where p is the number of punctures on Σ. There is a natural involution θ on T (G) which is defined as follows.

Suppose that an element [ρ] ∈ T (G) is given by a quasiconformal deformation f : \overline{ℂ} → \overline{ℂ} with the Beltrami coefficient μ. Define a new Beltrami differential ν on \overline{ℂ} by the formula:

ν(z) = \overline{μ}(z̄)

where we assume that Λ(G) is the real line {Im(z) = 0} ∪ {∞}. Then direct calculation shows that ν is invariant under G and thus corresponds to an element θ([ρ]) ∈ T (G). In terms of the decomposition T (G) = T (Σ) × T (Σ̄) one can describe θ as θ(τ_{1}, τ_{2}) = (τ_{2}, τ_{1}), where we identify the surfaces Σ, Σ̄ using orientation reversing isometry j induced by the complex conjugation. Therefore, an element τ = [ρ] = β(τ_{1}, τ_{2}) is a Fuchsian representation iff τ = θ(τ). For instance, suppose that ϕ, ψ ∈ Out_{a}(G) are two different outer automorphisms. They project to two quasiconformal homeomorphisms h_{1}, h_{2} of the surfaces Σ, Σ̄. Thus, (h_{1}, h_{2}) defines an element τ of of T (Σ) × T (Σ̄) as ([Σ, h_{1} : Σ → Σ], [Σ̄, h_{2} : Σ̄ → Σ̄]). The point τ is not fixed under the involution θ since jf_{2}j : Σ → Σ is not homotopic to f_{1} : Σ → Σ. Therefore, the representation [ρ] = β(τ) is not Fuchsian. An example of this situation is when ϕ = ψ^{−1} is an outer automorphism of infinite order.

Corollary 9.34 Suppose that G is a geometrically finite Kleinian group such that the manifold \overline{M}(G) has incompressible boundary. Let F_{1}, ..., F_{k} be the collection of subgroups in G corresponding to components of Ω(G)/G. Then T (G) ↪ T (F_{1}) × ... × T (F_{k}).

9.10 Distortion of the translational length under quasiconformal conjugation

Theorem 9.35 Suppose that f : \overline{ℂ} → \overline{ℂ} is a K-quasiconformal homeomorphism which conjugates two loxodromic elements g_{1}, g_{2} with the translational lengths ℓ(g_{1}), ℓ(g_{2}). Then

K^{−1} ≤ ℓ(g_{2})/ℓ(g_{1}) ≤ K  (35)

Proof: The quotient Ω(⟨g_{j}⟩)/⟨g_{j}⟩ is a torus T_{j}. Up to conformal conjugation, the elements g_{j} have the form z ↦ λ_{j}z, where |λ_{j}| = exp(ℓ(g_{j})). Consider the universal covering

exp : ℂ → ℂ^{∗} = Ω(⟨g_{j}⟩)

Then the preimages of the fundamental rings {1 ≤ |z| ≤ |λ_{j}|} of ⟨g_{j}⟩ under the exponential map are the vertical strips S_{j} = {0 ≤ Im(z) ≤ log(|λ_{j}|)}.

The homeomorphism f descends to a quasiconformal homeomorphism q : T_{1} → T_{2}. The coefficient of quasiconformality of q is not less than the coefficient of quasiconformality K(h) of an extremal affine map h : T_{1} → T_{2} which is homotopic to q. A lift of h to the universal cover ℂ is an affine map A which sends S_{1} to the strip S_{2}. Therefore we can the calculations in [1, Section ??], to conclude that the coefficient of quasiconformality K(A) = K(h) is at least log(|λ_{2}|)/ log(|λ_{1}|) = ℓ(g_{2})/ℓ(g_{1}). Thus, K = K(f) ≥ ℓ(g_{2})/ℓ(g_{1}). The consideration of the inverse map f^{−1} which has the same coefficient of quasiconformality K yields the estimate. □

9.11 Proof of the Ahlfors Recurrence Theorem

Proof of Theorem 6.80. Let’s use quasiconformal deformations of Kleinian groups to prove the Ahlfors Recurrence Theorem. Recall that G is a finitely generated discrete subgroup of Isom(ℍ^{3}). Suppose that there exists a subset A ⊂ Λ(G) of nonzero measure such that mes(gA ∩ A) ≠ 0 only for finite number of elements g ∈ G. Then, after taking a finite subdivision of A, we can assume that this finite set consist only of 1. Let μ_{A} ∈ L^{∞}(A, ℂ) such that ∥μ_{A}∥ < 1; we denote the space of such μ_{A} by Belt(A). Extend μ_{A} by zero to \overline{ℂ} − G(A) and by g_{∗}(μ_{A}) on each set gA, where g ∈ G. The result of this extension is a G-invariant Beltrami differential μ, ∥μ∥ < 1 and the map μ_{A} ↦ μ is linear. Then there exists a quasiconformal map f = f^{μ} normalized as f(0, 1, ∞) = (0, 1, ∞) which is the solution of Beltrami equation ¯∂f = μ∂f. The map f conjugates G to a discrete group G′ = f^{−1}Gf ⊂ Isom(ℍ^{3}). Therefore, like in the proof of Bers’ theorem, we have a continuous embedding β : Belt(A) → R(G, Isom(ℍ^{3})). (The fact that β is injective in this case follows from the fact that a quasiconformal conjugation of a discrete Moebius group is never induced by a conjugation in Isom(ℍ^{3}) unless the restriction of the quasiconformal homeomorphism to the limit set is conformal.) However, the space Belt(A) is infinite– dimensional, and the representation variety R(G, Isom(ℍ^{3})) has finite dimension since G is finitely generated. This contradiction concluded the proof of Theorem 6.80. □

One may wonder why do we consider the Ahlfors Recurrence Theorem as a companion of the Finiteness Theorem. The reason is that their proofs are very similar. For instance, suppose that G is a torsion-free finitely generated Kleinian group such that S = Ω(G)/G contains infinitely many components each of which is different from the triply punctured sphere. Then T (S) is infinite dimensional which contradicts the fact that R(G, Isom(ℍ^{3})) has finite dimension. The same argument works if one of the components of S does not have a finitely-generated fundamental group. It takes a bit more effort to extend this argument to the case when S contains components of infinite conformal (but finite topological) type as well as to handle the case of infinitely many triply punctured spheres in the quotient. (The latter was a gap in the original Ahlfors’ proof.)

9.12 A generalization

In the section we consider a class of Kleinian groups Γ in Isom(ℍ^{3}) which have torsion. We make the following list of assumptions:

• Γ is geometrically finite;

• Ω(Γ) consists of simply-connected components Ω_{j};

• The stabilizer Γ_{j} of each component Ω_{j} has a fundamental domain Φ_{j} ⊂ Ω_{j} which is either a quadrilateral or a triangle, whose edges are circular arcs tangent at the vertices;

• Each Γ_{j} is generated by inversions in the edges of Φ_{j}.

The class B of Kleinian groups which satisfy these conditions looks rather random, but it will be quite useful in the proof of Brooks’ Theorem 19.2. Let q = q(Γ) denote the number of components of Ω(Γ)/Γ which are quadrilaterals .

First we look at the quotient Ω_{j}/Γ_{j} = Φ_{j}. If we double Φ_{j} along its boundary (which does not include vertices), the result is a 3-punctured or 4-punctured sphere. Let S(4) denote a 4-punctured sphere which appears as the double of one of these quadrilaterals . Then S(4) admits an anticonformal automorphism J which fixes the union of edges of this quadrilateral . Thinking about S(4) as the usual extended complex plane \overline{ℂ} with 4 points removed, we conclude that J acts as an inversion in \overline{ℂ} which fixes these punctures, hence they must belong to a common round circle in \overline{ℂ}. The opposite is true as well, given 4 points z_{0}, z_{1}, z, z_{∞} in \overline{ℂ} which lie on a common round circle \overline{ℝ} and a reflection J which fixes this circle, the quotient [\overline{ℂ}−\{z_{0}, z_{1}, z, z_{∞}\}]/J is quasiconformal equivalent to each of the “ideal” quadrilaterals Φ_{j} that we have started with. The group of Moebius transformations acts transitively on the set of triples of distinct points. Thus we can assume z_{0} = 0, z_{1} = 1, z_{∞} = ∞ and z is a point on the ray (1, ∞). The real parameter z gives a parameterization to the space of conformal structures on the ideal quadrilateral Φ_{j}. Alternatively we can take the conformal modulus of the quadrilateral Φ_{j} with marked vertices. Let R_{x} be the rectangle in ℂ with the vertices {0, x, x + i, i}. Then find a Riemann mapping

f_{z} : Φ_{j} → R_{x}

The number x = m(Φ_{j}) ∈ (0, +∞) is the conformal modulus of Φ_{j}, which uniquely determines this quadrilateral up to conformal equivalence.

Take the subgroup Γ^{0} of index 2 in Γ, which preserves orientation in ℍ^{3}. Then Ω(Γ^{0})/Γ^{0} is a collection of 3- and 4-punctured spheres. Now we consider the Teichmüller spaces T (Γ), T (Γ^{0}). According to Bers’ theorem, the Teichmüller space T (Γ^{0}) is canonically isomorphic to the product:

∏_{j=1}^{q} T (S(4))

since 3-punctured spheres are conformally rigid. Quasiconformal deformations

f : \overline{ℂ} → \overline{ℂ}, fΓ^{0}f^{−1} ⊂ P SL(2, ℂ)

which extend to quasiconformal deformations of the group Γ are exactly those whose Beltrami differentials in components Ω_{j} are invariant under inversions from Γ_{j}. This means that they are coming from quasiconformal deformations of ideal quadrilaterals Φ_{j}. Hence we have a natural homeomorphism

β : ∏_{j=1}^{q} ℝ_{+} → T (Γ)

so that β^{−1} : Γ ↦ (m(Φ_{1}), ..., m(Φ_{q})).

9.13 Totally degenerate groups

Definition 9.36 Suppose that Γ is a finitely generated nonelementary Kleinian subgroup in Isom(ℍ^{3}). The group Γ is called totally degenerate if its limit set is connected and simply connected.

Existence of such groups was established by Bers in 1960-s. However, so far nobody was able to give a constructive example of such group. (By this I mean: to give matrices of generators whose entries belong to ℚ̄). Curt McMullen in [141] gives an algorithm which produces approximates to matrices of generators for a totally degenerate group. The question about matrices with entries from ℚ̄ can look strange, but the fact that we do not even know if totally degenerate groups defined over ℚ̄ exist (I believe that they do), means that we still lack in understanding of Kleinian groups and their deformation spaces.

Below I describe an indirect construction of totally degenerate groups which is due to Lipman Bers. Fix a Fuchsian group F which is isomorphic to the fundamental group of a closed oriented hyperbolic surface S. Let ρ_{n} : F → F_{n} ⊂ P SL(2, ℂ) be a sequence of discrete and faithful and type-preserving representations. Suppose that there exists a limit ρ = lim_{n→∞} ρ_{n} and ρ(F) = Γ has no parabolic elements.

Theorem 9.37 The group Γ constructed above satisfies one of the following three mutually exclusive conditions:

(a) Γ is quasifuchsian,

(b) Γ is totally degenerate,

(c) the domain of discontinuity of Γ is empty.

Proof: Suppose that Ω(Γ) ≠ ∅, D ⊂ Ω(Γ) is a component with the stabilizer H. The Ahlfors’ finiteness theorem implies that H is finitely generated. If D is not simply–connected, then the surface D/H ⊂ ∂\overline{M}(Γ) is compressible in the manifold \overline{M}(Γ) = ℍ^{3} ∪Ω(Γ)/Γ. Therefore there exists an embedded disk X ⊂ \overline{M}(Γ) such that ∂X ⊂ D/H is homotopically nontrivial. This implies that Γ = π_{1}\overline{M}(Γ) splits as a nontrivial free product, which contradicts the assumption that Γ ≅ F ≅ π_{1}(S). So, all components of Ω(Γ) are simply–connected. Suppose that |Γ : H| = ∞. Then H ≅ π_{1}(D/H) is a finitely generated free group, which means that D/H has punctures. Hence H has parabolic elements. This contradiction proves that stabilizers of all components of Ω(Γ) have finite index in Γ. Suppose that Ω(Γ) is not connected (i.e. Γ is not a totally degenerate group). Then Γ has a finite–index subgroup K which stabilizes all components of Ω(Γ) = Ω(K). For each component D ⊂ Ω(K) the inclusion D/K ↪ \overline{M}(K) is a homotopy–equivalence. Thus H_{2}(\overline{M}(K), ℤ) ≅ ℤ. Therefore \overline{M}(K) is compact and has exactly two boundary components D_{1}/K, D_{2}/K. In particular, the group K is geometrically finite and has no parabolic elements. According to Theorem 9.15, the group K is quasifuchsian. Hence Γ is also quasifuchsian. □

It’s easy to see that any totally degenerate group is not geometrically finite. Indeed, suppose that Γ is totally degenerate and has no parabolic elements. If Γ would be geometrically finite, then Λ(Γ) would be a topological circle which contradicts the definition (Ω(Γ) must be connected and simply-connected). If Γ has parabolic elements and is geometrically finite, then each cusp in M(Γ) must correspond to a pair of punctures on the quotient surface S(Γ) = Ω(Γ)/Γ. Thus, two distinct punctures on S(Γ) corresponds to the same conjugacy class of parabolic elements in Γ and the epimorphism π_{1}(S(Γ)) → Γ is injective. This implies that S(Γ) is conformally equivalent to ℂ^{∗}, hence Γ is elementary.

Historically this was the first example of a geometrically infinite Kleinian group which is finitely generated (due to L. Greenberg who noticed this fact in 1967).

A nonconstructive example of a totally degenerate group.

Start with the Fuchsian group F which corresponds to a pair of points (σ, \overline{σ}) ∈ T (S) × T (\overline{S}) = T (F), where S is a closed hyperbolic orientable surface. Pick an orientation– preserving aperiodic homeomorphism h : S → S. This means that for every nontrivial loop γ ⊂ S and any n > 0 the loops γ and h^{n}(γ) are not freely homotopic. Existence of such maps will be discussed in Section 12.

Consider a sequence of points ϕ_{n} = (h^{n}σ, \overline{σ}) ∈ T (S) × T (\overline{S}), where the homeomorphism h^{n} acts as an element of the modular group on T (S). The sequence h^{n}σ is not relatively compact in T (S) since h_{∗} : F → F has infinite order and the modular group acts discretely on the Teichmüller space. The representation ϕ_{n} : F → F_{n} is induced by quasiconformal conjugation via a quasiconformal homeomorphism f_{n}, which is conformal in the unit disk Δ. Therefore f_{n}|_{Δ} is subconvergent and the limit f_{∞} is a nonconstant holomorphic map. The Schwarzian derivative {f_{n}, z} defines a quadratic differential which is F –invariant. By continuity {f_{∞}, z} is also F –invariant. Thus, f_{∞} induces a representation ϕ of F into P SL(2, ℂ). The representation ϕ is the limit of ϕ_{n}. Actually one can prove that the whole sequence ϕ_{n} is convergent.

The representation ϕ can not be quasifuchsian since in T (F) algebraic convergence implies convergence in T (S) × T (\overline{S}). The group ϕ(F) has nonempty discontinuity set which contains f_{∞}(Δ). One can prove that ϕ(F) has no parabolic elements (see Section 12). Thus, Theorem 9.37 forces ϕ(F) to be totally degenerate.

9.14 Algebraic topology versus geometric topology

Fix a finitely generated group F.

Definition 9.38 A sequence of representations ρ_{n} : F → P SL(2, ℂ) is convergent strongly to a representation ρ iff ρ_{n} is convergent to ρ algebraically and the sequence ρ_{n}(F) is convergent to ρ(F) geometrically.

As we saw, algebraic convergence does not necessarily imply strong convergence. Let S be an oriented hyperbolic surface of finite area, uniformized by a Fuchisan group F ⊂ P SL(2, ℝ).

Theorem 9.39 (W. Thurston, see [192] and the paper of Canary and Minsky [36] for careful and self-contained treatment.) Suppose ρ_{n} : F → F_{n} ⊂ P SL(2, ℂ) is a sequence of faithful quasi-Fuchsian representations so that lim_{n→∞} ρ_{n} = ρ and ρ : F → Γ is a type preserving isomorphism. Then

1. ρ_{n} is convergent strongly to ρ;

2. Moreover, if Γ has a component of domain of discontinuity Ω^{0} and Ω_{n}^{0} denotes the sequence of components of Ω(F_{n}) which are convergent to Ω^{0}, then Ω_{n}^{0}/F_{n} is convergent to Ω^{0}/Γ as a sequence of points in the Teichmüller space T (S).

A complete proof of this theorem would require another twenty pages long detour and I will omit proof of the assertion (1). See [192, Chapter 9] and the paper by K.-I. Ohshika [158]. I will present a proof of the Assertion (2) assuming for simplicity that S is a compact surface. In Section 13.6 we will prove the Assertion (1) under the assumption that the domain of discontinuity of Γ is empty.

Proof: If the group Γ is quasi-Fuchsian, then it must belong to T (F). Therefore the conclusion follows from the Bers’ correspondence: Teichmüller topology on T (F) is equivalent to the algebraic topology.

Now suppose that Ω(Γ) consists of a single component Ω_{0}, α is the marked complex projective structure on Ω_{0}/Γ. It follows from Theorem 8.2 that there is a sequence α_{n} ∈ MCP (S) which is convergent to α and α_{n} have monodromy ρ_{n}. The developing maps d_{n} : ℍ^{2} → \overline{ℂ} of α_{n} are convergent to the developing map d of α uniformly on compacts in C^{1} topology. Let D be a compact fundamental domain for the action of F on ℍ^{2}. Since d is a quasiconformal homeomorphism on its image, d(D) is a fundamental domain for the action of Γ on Ω_{0}. Hence, Hausdorff convergence of limit sets of ρ_{n}(F) to the limit set of Γ implies that d_{n}(D) ⊂ Ω_{n}^{0} for large n. Therefore the image of the developing map d_{n}(ℍ^{2}) lies inside Ω_{n}^{0}. The map d_{n} projects to a local homeomorphism

f_{n} : S = ℍ^{2}/F → Ω_{n}^{0}/F_{n} = S_{n}

The surface S_{n} is homeomorphic to S, thus the covering f_{n} is a quasiconformal homeomorphism. This implies that d_{n} : ℍ^{2} → Ω_{n}^{0} are quasiconformal homeomorphisms which are convergent to d uniformly on compacts in C^{1} topology. □

9.15 Geometric profile of the algebraic convergence

In this section we analyse what happens to Scott’s compact cores of sequences of hyperbolic manifolds, whose fundamental groups are convergent algebraically. This analisys will be used in Section 17.2.3.

Suppose that Γ is a nonelementary geometrically finite torsion-free subgroup in Isom(ℍ^{3}), M = ℍ^{3}/Γ and M̄ has incompressible boundary S(Γ). Consider a sequence of discrete and faithful representations ρ_{n} : Γ → Γ_{n} ⊂ Isom(ℍ^{3}) which is convergent algebraically to a homomorphism ρ_{∞} : Γ → Γ_{∞}. Let M_{n} := M(Γ_{n}) = ℍ^{3}/Γ_{n}. We assume that each ρ_{n} is induced by a quasiconformal homeomorphism h̃_{n} : \overline{ℍ^{3}} → \overline{ℍ^{3}}, in particular each group Γ_{n} is geometrically finite and ρ_{n} are type-preserving. Let h_{n} : M → M_{n} denote the projection of this homeomorphism. Pick a number ϵ < μ_{3}. We have a sequence of truncated hyperbolic manifolds M_{n}^{0} := M(Γ_{n})_{ϵ}^{0}, M(Γ_{∞})_{ϵ}^{0} (obtained by removing Margulis cusps). The manifold M_{n}^{0} has a compact core M_{n}^{c} (see Theorem 6.82). Our goal is to understand what happens to compact cores M_{n}^{c} ⊂ M(Γ_{n})_{ϵ}^{0} as n → ∞. Keep in mind, that we can not choose compact cores M_{n}^{c} which are convergent to the compact core of M(Γ_{∞}) in (say) topology induced by Hausdorff distance between compact metric spaces. For instance, Anderson and Canary [7] constructed examples when the compact core of M(Γ_{∞}) is not homeomorphic to M^{c}. Nevertheless, we will try to extract some geometric information about compact cores.

Let ξ_{n} := ρ_{n}∘ρ^{−1} : Γ_{∞} → Γ_{n}, this sequence of homomorphisms is algebraically converges to the identity representation. The manifold M_{∞} could have more cusps then the manifolds M_{n} = M(Γ_{n}). So, we consider the union C_{∞} of those cusps in the ϵ-thin part of M_{∞}, which correspond to parabolic elements under the isomorphism ρ_{∞} : Γ → Γ_{∞} of the fundamental groups. Let M_{∞}^{1} denote M_{∞} − C_{∞} and ∂_{P} M_{∞}^{1} denote the intersection of ∂M_{∞}^{1} and cl(C_{∞}), ∂_{P} M_{∞}^{c} := ∂M_{∞}^{c} ∩ cl(C_{∞}); similarly the boundary ∂_{P} M_{n}^{0} is the intersection of M_{n}^{0} with the closure of the union of Margulis cusps in M_{n}.

Take a relative Scott compact core M_{∞}^{c} of the manifold M_{∞}^{1}. Let M̃_{∞}^{c} denote the lift of this compact core to the universal cover ℍ^{3} of the manifold M_{∞} := M(Γ_{∞}).

Lemma 9.40 There exists a sequence of ξ_{n}-equivariant (1 + 1/n)-Lipschitz maps λ̃_{n} : M̃_{∞}^{c} → ℍ^{3} which are C^{1}-convergent to the identity uniformly on compacts in M̃_{∞}^{c}. These maps project to immersions λ_{n} : M_{∞}^{c} → M_{n}^{0}, which are injective at ∂_{P} M_{∞}^{c} and send ∂_{P} M_{∞}^{c} to ∂_{P} M_{n}^{0}.

Proof: Repeat our arguments from the proof of Theorem 8.2. □

Lemma 9.41 Suppose that the convergence ξ_{n} → id is strong. Then the maps λ_{n} are injective for all sufficiently large n.

Proof: Choose a fundamental domain D for the action of Γ_{∞} on M̃_{∞}^{c} (it is a compact domain) and let U be a the union of D and the images of D under face-pairing transformations (this domain is still compact). Since λ̃_{n} are convergent to the identity on compacts in M̃_{∞}^{c}, the restrictions of these maps to U are injective for large n. The only way the map λ_{n} can fail to be injective in this situation is that for each n there exists γ_{n} ∈ Γ_{n} so that γ_{n}λ̃_{n}(D) ∩ λ̃_{n}(D) ≠ ∅ and ξ_{n}^{−1}(γ_{n}) is not one of the face-pairing transformations in Γ_{∞}. If all ξ_{n}^{−1}(γ_{n}) belong to a finite subset in Γ_{∞}, then after taking a limit n → ∞ we get a contradiction with the fact that D is a fundamental domain. If ξ_{n}^{−1}(γ_{n}) do not belong to a finite subset of Γ_{∞}, they are still convergent to an element γ_{∞} ∈ Isom(ℍ^{3}), however this limit is not in the group Γ_{∞}. This contradicts the assumption that the convergence is strong. □

Let M_{n}^{d} := λ_{n}(M_{∞}^{c}). Note that for large n, the diameter of M_{n}^{d} is at most 2+diam(M_{∞}^{c}). The manifolds M_{n}^{0} admit compactifications to manifolds with boundary (since they are geometrically finite). Hence, after taking sufficiently large Scott compact cores M_{n}^{c} of M_{n}^{0}, we can assume that M_{n}^{d} ⊂ M_{n}^{c}.

Now choose a component S of ∂_{0}M^{c}. We know that h_{n} : S → h_{n}(S) ⊂ M_{n} is a homeomorphic embedding. By adjusting h_{n} we can assume that h_{n}(S) is a component of ∂_{0}M_{n}^{c}. This surface is far away from the compact M_{n}^{d}. However we also have a continuous map h_{∞} : S → M_{∞}^{c} which induces the embedding ρ_{n} : π_{1}(S) → π_{1}(M_{∞}^{c}). (Note that this map is in no sense a limit if h_{n}’s). Since ρ_{∞} at least sends parabolic elements to parabolic we conclude that h_{∞} can be chosen so that h_{∞} : ∂S → ∂_{P} M_{∞}^{c} is injective.

Take the composition λ_{n} ∘ h_{∞} : S → M_{n}^{d}. This map is properly homotopic to h_{n} : S → h_{n}(S) as a map of pairs (S, ∂S) → (M_{n}^{c}, ∂_{0}M_{n}^{c}). Hence by Theorem 2.30 there is a

Figure 22: Geometric profile of the algebraic convergence.

properly homotopic embedding e_{n} : (S, ∂S) → (M_{n}^{c}, ∂_{P} M_{n}^{c}) whose image is contained in the 1-neighborhood of M_{n}^{d}. Note that the embeddings e_{n} and h_{n}|_{S} are homotopic as maps of pairs and their images are disjoint. Therefore they bound a product region S × I in M_{n}^{c}. See Figure 22.

We repeat this procedure for other components S′ of ∂_{0}M^{c} so that images e_{n}(S), e_{n}(S′) are disjoint.

Now consider a special case: π_{1}(S) = Γ. The manifold M_{∞}^{1} is naturally homeomorphic to the product S × ℝ via a homeomorphism f_{∞} : M^{0} ≅ S × ℝ → M_{∞}^{1}. We identify S × {0} ⊂ M_{∞}^{1} with the surface f_{∞}(S). Take a sufficiently big compact core M_{∞}^{c} which contains the 4d-neighborhood of f_{∞}(S), where d = diam(f_{∞}(S)). Then for large n we have: λ_{n}(M_{∞}^{c}) ⊃ e_{n}(S). Recall that λ_{n} is a local diffeomorphism in the interior of M_{∞}^{c}, thus ∂(λ_{n}(M_{∞}^{c})) ⊂ λ_{n}(∂M_{∞}^{c}). This implies that λ_{n}-images of two components of ∂_{0}(M_{∞}^{c}) lie by different sides of the surface e_{n}(S). We conclude that there is a natural 1-1 correspondence between the two topological ends of M_{∞}^{1} and two topological ends of M_{n}^{0} for large n.

10 Ultralimits of Metric Spaces

Let (X_{i}) be a sequence of metric spaces. One can describe the limiting behavior of the sequence (X_{i}) by studying limits of sequences of finite subsets Y_{i} ⊂ X_{i}. Ultrafilters are an efficient technical device for simultaneously taking limits of all such sequences of subspaces and putting them together to form one object, namely an ultralimit of (X_{i}) (see [109, 110, 108, 48] for application of ultralimits to study of quasi-isometries of metric spaces). We discuss this concept following [83] and [109].

10.1 Ultrafilters

Let I be an infinite set, S is a collection of subsets of I. A filter based on S is a nonempty family ω of subsets of I with the properties:

• ∅ ∉ ω.

• If A ∈ ω and A ⊂ B, then B ∈ ω.

• If A_{1}, …, A_{n} ∈ ω, then A_{1} ∩ … ∩ A_{n} ∈ ω.

If S consists of all subsets of I we will say that ω is a filter on I. Subsets A ⊂ I which belong to a filter ω are called ω-large. We say that a property (P) holds for ω-all i, if (P) is satisfied for all i in some ω-large set. An ultrafilter is a maximal filter. The maximality condition can be rephrased as: for every decomposition I = A_{1} ∪ … ∪ A_{n} of I into finitely many disjoint subsets, the ultrafilter contains exactly one of these subsets.

For example, for every i ∈ I, we have the principal ultrafilter δ_{i} defined as δ_{i} := {A ⊂ I | i ∈ A}. An ultrafilter is principal if and only if it contains a finite subset. The interesting ultrafilters are of course the non-principal ones. They cannot be described explicitly but exist by Zorn’s lemma: every filter is contained in an ultrafilter. Let Z be the Zariski filter which consists of complements to finite subsets in I. An ultrafilter is a nonprincipal ultrafilter, if and only if it contains Z. For us is not important how ultrafilters look like, but rather how they work: An ultrafilter ω on I assigns a “limit” to every function f : I → Y with values in a compact space Y. Namely,

ω-lim f = ω-lim_{i} f(i) ∈ Y

is defined to be the unique point y ∈ Y with the property that for every neighborhood U of y the preimage f^{−1}U is “ω-large”. To see existence of a limit, assume that there is no point y ∈ Y with this property. Then each point z ∈ Y possesses a neighborhood U_{z} such that f^{−1}U_{z} ∉ ω. By compactness, we can cover Y with finitely many of these neighborhoods. It follows that I ∉ ω. This contradicts the definition of a filter. Uniqueness of the point y follows, because Y is Hausdorff. Note that if y is an accumulation point of {f(i)}_{i∈I} then there is a non-principal ultrafilter ω with ω-lim f = y, namely an ultrafilter containing the pullback of the neighborhood basis of y.

10.2 Ultralimits of metric spaces

Let (X_{i})_{i∈I} be a family of metric spaces parameterized by an infinite set I. For an ultrafilter ω on I we define the ultralimit

X_{ω} = ω-lim_{i} X_{i}

as follows. Let Π_{i}X_{i} be the product of the spaces X_{i}, i.e. it is the space of sequences (x_{i})_{i∈I} with x_{i} ∈ X_{i}. The distance between two points (x_{i}), (y_{i}) ∈ Π_{i}X_{i} is given by

d_{ω}((x_{i}), (y_{i})) := ω-lim(i ↦ d_{X_{i}}(x_{i}, y_{i}))

where we take the ultralimit of the function i ↦ d_{X_{i}}(x_{i}, y_{i}) with values in the compact set [0, ∞]. The function d_{ω} is a pseudo-distance on Π_{i}X_{i} with values in [0, ∞]. Set

(X_{ω}, d_{ω}) := (Π_{i}X_{i}, d_{ω})/ ∼

where we identify points with zero d_{ω}-distance.

Example 10.1 Let X_{i} = Y for all i, where Y is a compact metric space. Then X_{ω} ≅ Y for all ultrafilters ω.

The concept of ultralimits extends the notion of Gromov-Hausdorff limits:

Proposition 10.2 Let (X_{i})_{i∈ℕ} be a sequence of compact metric spaces converging in the Gromov-Hausdorff topology to a compact metric space X. Then X_{ω} ≅ X for all nonprincipal ultrafilters ω.

Proof: Realize the Gromov-Hausdorff convergence in an ambient compact metric space Y, i.e. embed the X_{i} and X isometrically into Y such that the X_{i} converge to X with respect to the Hausdorff distance. Then there is a natural isometric embedding

X_{ω} = ω-lim_{i} X_{i} ⟶^{ι} ω-lim_{i} Y ≅ Y

Since ω is non-principal, the ω-limit is independent of any finite collection of X_{i}’s and we get:

ι(X_{ω}) ⊆ ⋂_{i_{0}} \overline{⋃_{i≥i_{0}} X_{i}} = X

On the other hand X ⊆ ι(X_{ω}), because ι((x_{i})) = x if (x_{i}) is a sequence with x_{i} ∈ X_{i} converging in Y to x ∈ X. Hence ι(X_{ω}) = X which proves the claim. □

If the spaces X_{i} do not have uniformly bounded diameter, then the ultralimit X_{ω} decomposes into (generically uncountably many) components consisting of points of mutually finite distance. We can pick out one of these components if the spaces X_{i} have basepoints x_{i}^{0}. The sequence (x_{i}^{0})_{i} defines a basepoint x_{ω}^{0} in X_{ω} and we set

X_{ω}^{0} := {x_{ω} ∈ X_{ω} | d_{ω}(x_{ω}, x_{ω}^{0}) < ∞}

Define the based ultralimit as

ω-lim_{i}(X_{i}, x_{i}^{0}) := (X_{ω}^{0}, x_{ω}^{0})

Example 10.3 For every locally compact space Y with a basepoint y_{0}, we have:

ω-lim_{i}(Y, y_{0}) ≅ (Y, y_{0})

We observe that some geometric properties pass to ultralimits:

Proposition 10.4 Let (X_{i}, x_{i}^{0})_{i∈I} be a sequence of based geodesic spaces and let ω be an ultrafilter. Then X_{ω}^{0} is a geodesic space.

If the X_{i} are CAT(κ)–spaces for some κ ≤ 0 then X_{ω}^{0} has the same upper curvature bound κ.

Proof: The ultralimit of geodesic segments in X_{i} is a geodesic segment in X_{ω}^{0}. Therefore X_{ω}^{0} is a geodesic space. To verify the second assertion it suffices to prove that any pair of points x_{ω} = (x_{i}) and y_{ω} = (y_{i}) in X_{ω}^{0} can be joined by a unique geodesic (then we approximate triangle Δ_{ω} in X_{ω}^{0} by triangles Δ_{i} in X_{i}, the ω-limit of comparison triangles for Δ_{i} is the comparison triangle for Δ_{ω}). Suppose that d_{ω}(x_{ω}, y_{ω}) = s + t where s, t ≥ 0. There are points z_{i} on the geodesic segments [x_{i}y_{i}] such that for s_{i} := d_{i}(x_{i}, z_{i}) and t_{i} := d_{i}(z_{i}, y_{i}) we have ω-lim s_{i} = s and ω-lim t_{i} = t. Hence z_{ω} := (z_{i}) satisfies d_{ω}(x_{ω}, z_{ω}) = s and d_{ω}(z_{ω}, y_{ω}) = t. Suppose that u_{ω} = (u_{i}) is another point with the same property. Consider in the model space M_{κ}^{2} comparison triangles Δ(x′_{i}, u′_{i}, y′_{i}) with the same sidelengths as Δ(x_{i}, u_{i}, y_{i}). Let z′_{i} be a division point on [x′_{i}y′_{i}] corresponding to z_{i} on [x_{i}y_{i}]. Since ω-lim(d_{i}(x_{i}, u_{i}) + d_{i}(u_{i}, y_{i}) − d_{i}(y_{i}, x_{i})) = 0, we have ω-lim d_{i}(u_{i}, z_{i}) ≤ ω-lim d_{M_{κ}^{2}}(u′_{i}, z′_{i}) = 0 and therefore u_{ω} = z_{ω}. Thus there is a unique point z_{ω} ∈ X_{ω} with d_{ω}(x_{ω}, z_{ω}) = s and d_{ω}(z_{ω}, y_{ω}) = t. □

Recall that a metric tree is a geodesic metric space where any geodesic triangle is a tripod.

Exercise 10.5 If T is a metric tree, −∞ < a < b < ∞ and f : [a, b] → T is a continuous embedding then the image of f is a geodesic segment in T. (Hint: use PL approximation of f to show that the image of f contains the geodesic segment connecting f(a) to f(b).)

Corollary 10.6 Let (X_{i})_{i∈ℕ} be a sequence of geodesic spaces with upper curvature bounds κ_{i} tending to −∞. Then for every non-principal ultrafilter ω each component of the ultralimit X_{ω} is a metric tree.

Proof: Let Δ_{i} be a sequence of geodesic triangles in X_{i} which ω-converges to a triangle Δ_{ω} in X_{ω}. Let \overline{Δ}_{i} ⊂ ℍ^{2}(κ_{i}) be the comparison triangles for Δ_{i}, then the radius r_{i} of the inscribed circle in \overline{Δ}_{i} tends to zero as i → ∞. Therefore the ω-limit of Δ_{i} is a tripod. □

Lemma 10.7 Let X be a CAT (−1)–space, k, c be positive constants, then there is a function θ = θ(k, c) such that for any (k, c)-quasi-isometric embedding f : [a, b] → X the Hausdorff distance between the image of f and the geodesic segment [f(a)f(b)] ⊂ X is at most θ.

Proof: Suppose that the assertion of lemma is false. Then there exists a sequence of (k, c)- quasi-isometric embeddings f_{n} : [−n, n] → X_{n} to CAT (−1)-spaces X_{n} such that

lim_{n→∞} d_{H}(f([−n, n]), [f(−n), f(n)]) = ∞

where d_{H} is the Hausdorff distance in X_{n}. Let d_{n} := d_{H}(f([−n, n]), [f(−n), f(n)]). Pick points t_{n} ∈ [−n, n] such that |d(t_{n}, [f(−n), f(n)]) − d_{n}| ≤ 1. Consider the sequence of pointed metric spaces (\frac{1}{d_{n}} X_{n}, f_{n}(t_{n})), (\frac{1}{d_{n}} [−n, n], t_{n}). It is clear that ω-lim n/d_{n} > 1/k > 0 (but this ultralimit could be infinite). Let (X_{ω}, x_{ω}) = ω-lim(\frac{1}{d_{n}} X_{n}, f_{n}(t_{n})) and (Y, y) := ω-lim(\frac{1}{d_{n}} [−n, n], t_{n}). The metric space Y is either nondegenerate segment in ℝ or a closed geodesic ray in ℝ or the whole real line. Note that the Hausdorff distance between the image of f_{n} in \frac{1}{d_{n}} X_{n} and [f_{n}(−n), f_{n}(n)] ⊂ \frac{1}{d_{n}} X_{n} is at most 1 + 1/d_{n}. Each map

f_{n} : \frac{1}{d_{n}} [−n, n] → \frac{1}{d_{n}} X_{n}

is a (k, c/n)-quasi-isometric embedding. Therefore the ultralimit

f_{ω} = ω-lim f_{n} : (Y, y) → (X_{ω}, x_{ω})

is a (k, 0)-quasi-isometric embedding, i.e. it is a k-bilipschitz map:

|t − t′|/k ≤ d(f_{ω}(t), f_{ω}(t′)) ≤ k|t − t′|

In particular this map is a continuous embedding. On the other hand, the sequence of geodesic segments [f_{n}(−n), f_{n}(n)] ⊂ \frac{1}{d_{n}} X_{n} also ω-converges to a nondegenerate geodesic γ ⊂ X_{ω}, this geodesic is either a finite geodesic segment or a geodesic ray or a complete geodesic. In any case the Hausdorff distance between the image L of f_{ω} and γ is exactly 1, it equals the distance between x_{ω} and γ which is realized as d(x_{ω}, z) = 1, z ∈ γ. I will consider the case when γ is a complete geodesic, the other two cases are similar and are left to the reader. Then Y = ℝ and by Exercise 10.5 the image L of the map f_{ω} is a complete geodesic in X_{ω} which is within Hausdorff distance 1 from the complete geodesic γ. This contradicts the fact that X_{ω} is a metric tree. □

10.3 The asymptotic cone of a metric space

Let X be a metric space and ω be a non-principal ultrafilter on ℕ. The asymptotic cone Cone_{ω}( X) of X is defined as the based ultralimit of rescaled copies of X:

Cone_{ω}( X) := X_{ω}^{0}, where (X_{ω}^{0}, x_{ω}^{0}) = ω-lim_{i}(\frac{1}{i} · X, x^{0})

The limit is independent of the chosen basepoint x^{0} ∈ X. The discussion in the previous section implies:

Proposition 10.8 1. Cone_{ω}( X × Y) = Cone_{ω}( X) × Cone_{ω}( Y).

2. Cone_{ω}ℝ^{n} ≅ ℝ^{n}.

3. The asymptotic cone of a geodesic space is a geodesic space.

4. The asymptotic cone of a CAT(0)-space is CAT(0).

5. The asymptotic cone of a space with a negative upper curvature bound is a metric tree.

Remark 10.9 Suppose that X admits a cocompact discrete action by a group of isometries. The problem of dependence of the topological type of Cone_{ω}X on the ultrafilter ω is open (see [83]).

To get an idea of the size of the asymptotic cone, note that in the most interesting cases it is homogeneous. We call a metric space X quasi-homogeneous if diam(X/Isom(X)) is finite.

Proposition 10.10 Let X be a quasi-homogeneous metric space. Then Cone_{ω}( X) is a homogeneous metric space for every non-principal ultrafilter ω.

Proof: The group of sequences of isometries Isom(X)^{ℕ} acts transitively on the ultralimit ω-lim_{i}(\frac{1}{i} · X) which contains Cone_{ω}( X) as a component. □

Lemma 10.11 Let X be a quasi-homogeneous CAT (−1)–space with uncountable number of ideal boundary points. Then for every nonprincipal ultrafilter ω the asymptotic cone Cone_{ω}( X) is a tree with uncountable branching.

Proof: Let x^{0} ∈ X be a basepoint and y, z ∈ ∂_{∞}X. Denote by γ the geodesic in X with the ideal endpoints z, y. Then Cone_{ω}([x^{0}, y)) and Cone_{ω}([x^{0}, z)) are geodesic rays in Cone_{ω}( X) emanating from x_{ω}^{0}. Their union is equal to the geodesic Cone_{ω}γ. This produces uncountably many rays in Cone_{ω}( X) so that any two of them have precisely the basepoint in common. The homogeneity of Cone_{ω}( X) implies the assertion. □

11 Actions of Groups on Trees

11.1 Basic definitions and properties

Recall that a metric tree T is a (nonempty) geodesic metric space where each geodesic triangle is a tripod. In particular, every two points in T are connected by unique geodesic segment. Vertices (or branchipoints )of T are the points x ∈ T such that there are at least three geodesic segments in T emanating from x whose interiors are disjoint. A nondegenerate tree is a tree which contains more than one point.

We shall consider isometric actions of groups G on trees T. The pair: a tree T and an action of G on T is called a G-tree. A G-tree is called minimal if there is no G–invariant subtree in T different from T. The action is called small if stabilizer of any nondegenerate arc is an almost nilpotent subgroup of G. Two G-trees are said to be isomorphic if there is a G-equivariant isometry between these trees.

For every element g ∈ G the action of G on T defines the translational length ℓ_{T} (g) as follows:

ℓ_{T} (g) = inf{d(x, gx) : x ∈ T }  (36)

Definition 11.1 An isometry g of T is called hyperbolic or axial if it has an invariant geodesic A (called the axis of g) in T, so that g acts on A as a translation, in which case

ℓ_{T} (g) = Length(A/⟨g⟩)

An isometry g of T is called elliptic if it has a fixed point on T.

Note that each axial element g has unique axis; if T′ ⊂ T is a g-invariant subtree of T then g is axial if and only if g|T′ is axial.

Lemma 11.2 (See [42].) Each isometry of T is either hyperbolic or elliptic.

Proof: If g ∈ Isom(T) has a fixed point in T then there is nothing to prove. So, suppose that the action of g on T is free. Pick a point x ∈ T, let α be the geodesic segment in T which connects x and g(x). Consider the segments gα, g^{−1}α. Let a denote the point of intersection between α, g^{−1}α which is the most distant from x; let b = g(a) and β ⊂ α be the geodesic subsegment between a, b. It follows that b ≠ a since g acts freely. Then the ⟨g⟩–orbit A of β is an isometric copy of the real line in T and it is invariant under ⟨g⟩. Suppose that z ≠ A, then

d(z, gz) = 2d(x, A) + d(a, b)  (37)

Therefore d(a, b) realizes the infimum in the definition of ℓ_{T} (g). □

Corollary 11.3 Suppose that g has zero translational length in T. Then g has a fixed point in T.

Lemma 11.4 (See [42]) Assume that the elements g, h, hg act on the tree T as elliptic isometries. Then the group generated by g and h has a global fixed point on T.

Proof: Suppose that fixed–point sets F_{g}, F_{h} of g, h are disjoint, these are closed convex subsets in T. Pick points x ∈ F_{g}, y ∈ F_{h} and connect them by geodesic segment [xy]. Let α = [pq] ⊂ [xy] be the smallest subsegment such that p ∈ F_{g}, q ∈ F_{h}. Then the path (α) ∗ (hα) = ω is geodesic in T. The segments ω and hg(ω) meet only in a single point h(p) which is one of the end–points of ω. Therefore the ⟨hg⟩–orbit of ω is a geodesic in T, which implies that hg has no fixed points in T. Contradiction. □

Corollary 11.5 Suppose that G is a finitely generated group which acts on a tree T without a global fixed point. Then G contains an axial element.

Proof: Suppose that G consists ofnly of elliptic elements. Then by the previous lemme, for F_{g} ∩ F_{h} ≠ ∅ for each pair of elements g, h ∈ G. Let x_{ij} ∈ F_{g_{i}} ∩ F_{g_{i}}, i, j = 1, 2, 3. The geodesic triangle with the vertices x_{12}, x_{23}, x_{31} is a tripod. It is clear that the central point of this tripod belongs to all three subsets F_{g_{i}}, i = 1, 2, 3. Continue inductively: suppose that each m-fold intersection

F_{g_{1}} ∩ ... ∩ F_{g_{m}}

is nonempty. Then pick points:

a ∈ F_{g_{1}} ∩ ... ∩ F_{g_{m}}, b ∈ F_{g_{2}} ∩ ... ∩ F_{g_{m+1}}, c ∈ F_{g_{3}} ∩ ... ∩ F_{g_{m+1}} ∩ F_{g_{1}}

Then the centraal point of the tripod with the vertices a, b, c is contained in each fixed-point set F_{g_{i}}, i = 1, ..., m + 1. It follows that each (m + 1)-fold intersections of fixed-point sets of elements of G is nonempty. Hence G has a global fixed point in T. □

Corollary 11.6 Let T be a G-tree so that G is finitely generated and does not fix a point in T. Then the minimal G-invariant subtree in T is the union of axes of hyperbolic elements of G.

Proof: The group G contains axial isometries of T. The union of axes of axial elements of G is a subtree T′ in T. It is clear that T′ is G-invariant and minimal. □

Lemma 11.7 Suppose that g is an isometry of a metric tree T such that there is a nondegenerate arc J ⊂ T on which g induces a nontrivial orientation-preserving partial isometry g : s → s′, s, s′ ⊂ J. Then g is axial and its axis contains s ∪ s′.

Proof: Let m, m′ be the mid-points of s and s′ = g(s). Connect m, m′ by the geodesic segment I, it is necessarily contained in J. Then our assumptions imply that gI ∩I = {m′}, g^{−1}I ∩ I = {m}. Therefore the union

⋃_{n} g^{n}(I) = L

is a g-invariant geodesic in T. □

Lemma 11.8 Let T be a minimal G-tree and [xy] ⊂ T be a nondegenerate arc. Then there exists an axial element f ∈ G whose axis contains [xy].

Proof: The group G contains axial isometries of T; the union of axes of axial elements of G is a minimal G-invariant subtree T′ in T. Minimality of T implies that T′ = T. Hence we can assume that the points x, y belong to the axes A_{h}, A_{g} of certain elements h, g ∈ G. We can assume that y ∉ A_{h}, x ∉ A_{g}. Let A_{h} ∩ [xy] = [x, q], A_{g} ∩ [xy] = [py]. We can assume that q lies betweein x, h(x) and p lies between g^{−1}(y), y (otherwise we change g or/and h to its inverse).

Then for sufficiently large n we have: the composition g^{n}h^{n} induces an orientationpreserving partial isometry of the geodesic segment [h^{−n}(p)g^{n}(q)]. Hence by Lemma 11.7 the segment [h^{−n}(p)g^{n}(q)] is contained in the axis of g^{n}h^{n}. On the other hand, [xy] ⊂ [h^{−n}(p)g^{n}(q)] (see Figure 23) which concludes the proof. □

Theorem 11.9 (M. Culler, J. Morgan, [42].) Suppose that Γ is a finitely-generated group and T, R are minimal Γ-trees, so that the associated length functions

ℓ_{T} : Γ → ℝ, ℓ_{R} : Γ → ℝ

coincide. Then there exists an equivariant isometry between T and R.

Note that even in the case of a finitely generated free group G it is not enough to verify equality of the length functions on a finite subset of G. However, if G is the fundamental group of a closed surface, a minimal small G-tree is uniquely determined by the translational lengths of a finite number of elements of G, see Corollary 12.27.

11.2 Actions on simplicial trees, graphs of groups and graphs of spaces

The concept of simplicial tree is a refinement of metric tree: simplicial tree Γ is a 1dimensional simply-connected CW-complex. The 0-cells of Γ are called vertices and 1-cells are called edges. Every simplicial tree Γ can be given structure of a metric tree T where each edge is given the unit length. Note however that the set of vertices of T is apriori different from the set of vertices of Γ.

A metric tree T is called simplicial if it is isometric to the metric tree obtained from a simplicial tree by the above procedure. Equivalently, the set V of vertices of T is discrete and components of T − V have integer lengths. The procedure of conversion of simplicial to metric trees allows us to transfer notations and results from the theory of metric trees to the theory of simplicial trees. The main difference will be the notion of G-tree and isomorphism: the action of group G on simplicial tree must preserve the CW-complex structure, an isomorphism between simplicial G-trees is a G-equivariant isomorphism of CW-complexes.

A group G acts on a simplicial tree T without inversions if the G-stabilizer of each edge e ⊂ T fixes this edge pointwise.

Suppose that T is a minimal nontrivial simplicial G-tree, where a group G acts without inversions so that T/G is a finite graph. According to the Bass-Serre theory (see [184]) any such action of G on T corresponds to a graph of groups decomposition of G which can be though as an iterated HNN extension and amalgamated free product decomposition. Below is a very brief description of this decomposition.

Suppose that we have a finite graph Γ where each edge is oriented. We assume that each vertex v of Γ is assigned a vertex group G_{v} and each edge e is assigned an edge group G_{e}. Each inclusion v ↪ e of a vertex into edge (as the initial or the terminal vertex) corresponds to a monomomorphism h_{ev} : G_{e} → G_{v} The collection

(Γ, {G_{e}, G_{v}, h_{ev} : where e, v are edges and vertices of Γ})

is called the graph of groups L. The fundamental group π(L) is defined as follows.

There is a collection of injective homomorphisms ψ_{v} : G_{v} → π(L), ψ_{e} : G_{e} → π(L) which satisfies the following universal property:

Suppose that we are given a group H and a collection of homomorphisms ϕ_{v} : G_{v} → H, ϕ_{e} : G_{e} → H so that

ϕ_{v} ∘ h_{ev} = ϕ_{e}

for each inclusion v ↪ e. Then there is a homomorphism

θ : π(L) → H

so that

θ ∘ ψ_{v} = ϕ_{v}, θ ∘ ψ_{e} = ϕ_{e}

If G is a group isomorphic to π(L) then we say that G has a graph of groups decomposition. According to Bass-Serre theory of actions of groups on trees [184] each graph of groups decomposition of G gives rise to an action of G on a simplicial tree T so that each vertex stabilizer is conjugate to the image in G of one of the vertex groups G_{v} (which injects in G) and each edge stabilizer is also conjugate to the image of one of the vertex groups G_{v} (which injects in G). The graph Γ is the quotient T/G. The graph of groups decomposition is said to be trivial if the action of G on T has a global fixed point. Clearly each (nontrivial) graph of groups decomposition gives rise to a (nontrivial) decomposition of G as amalgamated free product or HNN-extension.

The concept of graph of groups appears naturally in relation to Seifert- Van Kampen theorem.

Definition 11.10 Suppose that we are given a finite graph Γ and each vertex v and edge e of Γ is assigned a connected CW-complex X_{v}, X_{e} (they are called the vertex and edge spaces) and each inclusion v ↪ e (as initial or terminal vertex) corresponds to a closed cellular embedding ϵ_{ev} : X_{e} ↪ X_{v}. We assume that images of all these embeddings are disjoint. Glue the spaces X_{v}, X_{e} using these embeddings. The result is a graph of spaces Δ.

Each graph of spaces (L, {X_{v}, X_{e}, ϵ_{ev}}) determines a graph of groups where G_{v} is the image of π_{1}(X_{v}) in π_{1}(X), G_{e} is the image of π_{1}(X_{e}), the maps G_{e} → G_{v} are induced by the inclusions ϵ_{ev}.

Remark 11.11 Note that we do not assume that π_{1}(X_{v}), π_{1}(X_{e}) inject in π_{1}(X).

The following is a version of Seifert- Van Kampen Theorem (cf. [134, Theorem 2.2]):

Theorem 11.12 The fundamental group of each graph of spaces is naturally isomorphic to the fundamental group of the corresponding graph of groups.

Below we illustrate the above decomposition in the following special case:

• G is the fundamental group of a compact paired 3-manifold (M, P);

• the action on T is elliptic with respect to P, i.e. the fundamental group of each component of P has a fixed vertex on T;

• the action of G on T is small.

Our discussion follows more general arguments of [181].

Let P = P_{1} ∪...∪P_{q}. The quotient S = T/G be a finite graph. Given the action of G on the simplicial tree T we construct a complex K as follows. For each vertex v of S we take an aspherical CW complex K_{v} with the fundamental group G_{v} which is the stabilizer of ṽ ∈ T in G. Suppose that vertices v, w ∈ T are connected by an edge e of T. In each G_{v}, G_{w} we have cyclic subgroups equal to the stabilizer G_{e} of e ⊂ T in G. Choose the subcomplexes K_{v,e}, K_{w,e} in K_{v}, K_{w} whose fundamental groups are copies of G_{e} (we can assume that these subcomplexes are simple loops). We can assume also that these subcomplexes are chosen so that each π_{1}(P_{j}) ⊂ π_{1}(K) is conjugate to the fundamental group of a subcomplex K(P_{j}) ⊂ K which is disjoint from any K_{v,e}. (Here we use ellipticity of the action on T.) Note that each G_{e} is either trivial or cyclic.

Connect K_{v,e}, K_{w,e} by a copy of K_{v,e} × I according to the isomorphisms π_{1}(K_{v,e}) ≅ G_{e} ≅ π_{1}(K_{w,e}). We repeat this for each vertex and edge of S. The result is a complex K whose fundamental group is G. Now we construct a P L homotopy–equivalence h : M → K assuming that it is in general position at all middle loops L_{e} = K_{v,e} × {1/2} of K_{v,e} × I. Therefore the preimage of every L_{e} is a compact properly embedded 2-submanifold in M which is disjoint from P. The image of the fundamental group of every component of h^{−1}(L_{e}) is conjugate into the corresponding subgroup G_{e} in G. Using Dehn Lemma we can deform h so that for each edge e:

• Every component of h^{−1}(L_{e}) is incompressible;

• Any annulus in h^{−1}(L_{e}) is essential;

• Any disk in h^{−1}(L_{e}) is compressing.

• There are no properly isotopic components

This implies Theorems 2.35, 2.36. Now we restrict to the case: M = Σ×I, where Σ is a compact hyperbolic surface, P = ∂S×I. Then each component of h^{−1}(L_{e}) is an essential “vertical” annulus: γ_{j} × I, where we can assume that each γ_{j} is a simple closed geodesic in Σ.

So, we get a decomposition of the surface Σ by a union σ of simple closed disjoint geodesics. Lift this decomposition in ℍ^{2} and let Γ be the tree dual to the corresponding decomposition of the hyperbolic plane. Stabilizer of every vertex of Γ in G is a free nonabelian group, which is the fundamental group of the corresponding component of Σ − σ, whose image under h : M → K is contained in the complement to the set of all middle loops in K. Therefore it is contained in the stabilizer of a vertex of T.

Stabilizer of each edge is a cyclic subgroup of G and for two distinct edges their stabilizers generate a free nonabelian subgroup. Now we construct a G–equivariant map q : Γ → T as follows. Every vertex v of Γ is stabilized by a (nonabelian) subgroup G_{v} of G, we shall send it to the fixed point of G_{v} on T (the last point exists by the argument above and is unique since the action of G on T is small). We extend this map to the set of edges of Γ by an affine map. The resulting map can be noninjective only if it folds the tree Γ at some vertex v ∈ Γ, i.e. for two adjacent segments e, d at v in Γ we have f(e) = f(d). This implies that the groups G_{d}, G_{v} will fix a common arc f(e) = f(d). However the group generated by G_{e}, G_{d} is nonabelian, which contradicts the assumption that the action of G on T was small. Therefore, T is isomorphic to Γ ⊂ ℍ^{2}.

The collection of geodesics σ ⊂ Σ is said to be dual to the action of G on T.

11.3 Equivariant Gromov–Hausdorff convergence

In this section we define the equivariant version of Gromov–Hausdorff topology for metric spaces that was introduced by F. Paulin in [164].

Suppose that X_{n} (n = 1, 2, ..., ∞) is a sequence of metric spaces and with actions of a finitely generated group G on X_{n}. Then

lim_{n→∞}[G ↷ X_{n}] = [G ↷ X_{∞}]

where G ↷ X_{∞} is an isometric action of G on a metric space X_{∞} if the following is satisfied:

For each number ϵ > 0, compact A_{∞} ⊂ X_{∞} and a finite subset K ⊂ G, there exists a number n_{0} < ∞ such that for every n > n_{0} the following is true. There exist a compact A_{n} in X_{n}, a relation R_{n} ⊂ A_{n} × A_{∞} which is onto both factors such that:

(a) for every x_{n}, x′_{n} ∈ A_{n} and x_{∞}, x′_{∞} ∈ A_{∞} which are related as x_{n}R_{n}x_{∞} , x′_{n}R_{n}x′_{∞}, we have: |d_{X_{n}}(x_{n}, x′_{n}) − d_{X_{∞}}(x_{∞}, x′_{∞})| ≤ ϵ (so R_{n} is an ϵ–approximate isometry between A_{n} and A_{∞});

(b) for every x_{n} ∈ A_{n}, x_{∞} ∈ A_{∞}, g ∈ K, such that g(x_{n}) ∈ A_{n}, and x_{n}R_{n}x_{∞}, it follows that gx_{∞} ∈ A_{∞} and g(x_{n})R_{n}gx_{∞}.

11.4 Limits of isometric actions on CAT (0)-spaces

There are several ways to compactify the space of (conjugacy classes of) discrete and faithful representations D(Γ, G) of a finitely-generated group Γ by actions of Γ on trees. First such compactification was introduced in by Culler and Shalen in [43] (in the case of G = SL(2, ℂ)), later it was generalized by Morgan [151] (in the case G = Isom(ℍ^{n})) and further more general geometric constructions of this compactification were found by Bestvina [19] and Paulin [164].

In this section, following [109], we will give yet another interpretation of this compactification using the language of ultralimits.

Take a finitely generated group Γ with a finite generating set G. Let X_{n} be a sequence of CAT (0)-spaces and ρ_{n} : Γ → Isom(X_{n}) be a sequence of representations. For x ∈ X_{n}, we denote by D_{n}(x) the diameter of the set ρ_{n}(G)(x). Set D_{n} := inf_{x∈X_{n}} D_{n}(x). We assume that the sequence (ρ_{n}) diverges in the sense that lim_{n→∞} D_{n} = ∞. Choose points x_{n} ∈ X_{n} such that D_{n}(x_{n}) ≤ D_{n} + 1/n. For any non-principal ultrafilter ω, there exists a natural isometric action ρ_{ω} of Γ on the ultralimit of rescaled spaces

(X_{ω}, x_{ω}) := ω-lim_{n}(D_{n}^{−1} · X_{n}, x_{n})

X_{ω} is a CAT(0)-space and the action ρ_{ω} has no global fixed point. If X_{n} are CAT (−1)- spaces, then X_{ω} is a metric Γ-tree.

11.5 Representation varieties and actions of groups on trees

Now we can discuss the relation between actions of groups on trees and sequences of discrete and faithful representations into Isom(ℍ^{n}). Suppose that Γ is a finitely generated group which is not almost Abelian, [ρ_{i}] ∈ D(Γ, Isom(ℍ^{n})) is a sequence of discrete and faithful representations which is not relatively compact. For each ρ_{i} define the number D_{i} and a point x_{i} ∈ ℍ^{n} as in the Section 11.4, where X_{i} = ℍ^{n} for all i.

Theorem 11.13 (J.Morgan [151], M.Bestvina [19], F.Paulin [164], [165]). Under the conditions above there exists a subsequence [ϕ_{j} := ρ_{i_{j}}] and a metric Γ-tree T such that the following is satisfied:

1.

lim_{i→∞}[Γ ↷ D_{j}^{−1} · ℍ^{n}] = [G ↷ T]

where convergence is the equivariant Gromov-Hausdorff convergence in the sense of Section 11.3.

2. The action of Γ on T is minimal, small and the tree T is nondegenerate;

3. lim_{j→∞} ℓ(ϕ_{j}(g))/D_{j} = ℓ_{T} (g);

4. T is the minimal Γ-invariant subtree in the tree

X_{ω} = ω-lim_{i}(D_{i}^{−1} · ℍ^{n}, x_{i})

for some choice of the ultrafilter ω.

Proof: We will present here only a proof of the last two assertions, the rest see in [151], [19], [164], [165].

First of all, since the sequence [ρ_{i}] is not relatively compact, it follows that we can choose a subsequence i_{j} so that lim_{j→∞} D_{i_{j}} = ∞ (otherwise we conjugate ρ_{i} to ρ′_{i} so that the sequence of points x_{i} is relatively compact).

Choose an ultrafilter which contains the subsequence i_{j}. Our goal is to show that action of Γ on the tree X_{ω} satisfies (3), then the Assertion (4) will follow from Theorem 11.9. Pick g ∈ Γ. Consider the case when ρ_{i}(g) is loxodromic for ω-all i. Let L_{i} denote the axis of ρ_{i}(g) and proj_{i} : ℍ^{n} → L_{i} be the nearest-point projection. There are two possible cases:

Case 1: ω-lim_{i} D_{i}^{−1}d(x_{i}, L_{i}) < ∞. Then ω-lim_{i} L_{i} = L_{ω} ⊂ X_{ω} is a g-invariant geodesic and the assertion is obviously true.

Case 2: ω-lim_{i} D_{i}^{−1}d(x_{i}, L_{i}) = ∞. If

ω-lim_{i} ℓ(ρ_{i}(g))/D_{i} > 0

then the exponential contraction property of proj_{i} (Lemma 4.4) implies that

d(x_{ω}, gx_{ω}) = ω-lim_{i} d(x_{i}, g(x_{i}))/D_{i} = ∞

which is impossible. Thus

ω-lim_{i} ℓ(ρ_{i}(g))/D_{i} = 0

Let us prove that g acts as an elliptic element on T. The ultralimit of the rescaled distance functions d(•, L_{i})/D_{i} is a Busemann function β on X_{ω}, which must be invariant under the action of g (compare [110]. Let ξ ∈ ∂_{∞}X_{ω} be the ideal boundary point corresponding to β, g(ξ) = ξ. If g is not elliptic, it must have an axis A on T. Connect ξ with A by a geodesic ray R which intersects A only at the origin. Then, since g acts on A as a translation, we conclude that gR ∩ R = ∅. Contradiction.

The cases when ρ_{i}(g) are elliptic or parabolic for ω-all i are similar to the Case 2 above, which finishes the proof of (4). □

11.6 Proper actions

Suppose that T is a complete metric space, and G is a finitely generated group which acts effectively and isometrically on T. Choose a finite system of generators of the group G:

g_{1}, ..., g_{r}

Then we get a function D : T → ℝ:

D(x) = max{d(g_{j}x, x), j = 1, ..., r}

Pick a base-point x_{0} ∈ T. The action of G on T is called proper if

lim_{d(x,x_{0})→∞} D(x, x_{0}) = ∞

(Thus the action is proper if the function D is “proper”, i.e. its sublevel sets are bounded.) Clearly properness of the action does not depend on the choice of generators. If G_{0} < G and the action of G_{0} on a G-tree T is proper, then the action of G is proper as well. An action of G on T is called axial if T ≅ ℝ. In this case G is almost abelian and a subgroup of index 2 in G acts on T by translations.

Theorem 11.14 Suppose that T is a nontrivial minimal G-tree. Then one of the following mutually exclusive cases occurs:

• either the action of G is proper;

• or the action of G is axial and G acts on ℝ by translations;

• or T is not a line and there exists an infinite ray R ⊂ T so that for each element h of the commutator subgroup [G, G] of G, there exists an infinite subray R(h) ⊂ R which is pointwise fixed by h. (We call this type of action unipotent.)

Proof: Since the action of G does not have a global fixed point, Lemma 11.4 implies that the group G contains a hyperbolic element g with the axis A. It follows from hyperbolicity of g that

d(gx, x) → ∞ as d(x, A) → ∞

Suppose that we have two hyperbolic elements g_{1}, g_{2} ∈ G whose axes intersect in a compact subset. Thus for any t > 0 the intersection

{x : d(x, g_{1}(x)) ≤ t} ∩ {x : d(x, g_{2}(x)) ≤ t}

is bounded. Hence the action of G_{0} := ⟨g_{1}, g_{2}⟩ on T is proper, which implies that the action of G is also proper. Now assume that the action of G is not proper. We conclude that axes of any two hyperbolic elements must intersect at least by an infinite ray. Then there exists an infinite ray R ⊂ T so that for any hyperbolic element g ∈ G the intersection between R and the axis A_{g} is an infinite subray. Let ξ ∈ ∂_{∞}T denote the ideal point corresponding to R. Then ξ is fixed by all hyperbolic elements of G. Let Γ be the normal subgroup of G generated by all hyperbolic element. Note that the fixed-point set for the action of Γ on ∂_{∞}T is invariant under the action of G. If this set contains more than one point, then Γ has an invariant geodesic L in T, which connects two of these points and which is the axis of all hyperbolic elements of G. Thus L is G-invariant and ℝ ≅ L = T, by minimality of T. This is the axial case, since the action is not proper, the group G must preserve the orientation on ℝ. In this case the assertion of Theorem is clearly valid.

Finally consider the case when Γ fixes exactly one point ξ on ∂_{∞}T. The point ξ is also fixed by the whole group G. We conclude that for any element h ∈ G there exists an infinite ray R(h) terminating in ξ so that R(h) is either pointwise fixed by h (if h is elliptic) or R(h) is contained in the axis of h (if h is hyperbolic). It is clear that for each element h ∈ [G, G] the ray R(h) can be chosen so that R(h) is pointwise fixed by h. □

Corollary 11.15 Suppose that under conditions of Theorem 11.14 the group G is not almost solvable and the action of G on T is small. Then this action is proper.

Proof: The only case we need to consider is the case of unipotent action. Since the action of G is small, the commutator subgroup of G must be almost nilpotent, hence G is almost solvable. □

Note that there are many examples of unipotent actions on trees. For instance, start with a 2-adic simplicial tree T (each vertex of T has valency 3 and all edges have unit length). Pick your favorite hyperbolic element h : T → T with the axis A. Since the group of automorphisms of T is so large, there exists an elliptic element g ∈ Aut(T) which fixes one half of A pointwise and sends another half to a distinct geodesic ray. Then take h′ := ghg^{−1}. The axis A′ of h′ equals g(A). Thus we let G ⊂ Aut(T) be the group generated by h, h′. To get a minimal action choose a minimal subtree T′ ⊂ T which is invariant under G. Clearly the action of G on T′ is unipotent.

12 Measured Laminations, Foliations and Actions of Surface Groups on Trees

In this chapter we discuss several essentially equivalent mathematical objects:

• Geodesic currents with zero self-intersection;

• measured foliations;

• measured geodesic laminations;

• trajectories of holomorphic quadratic differentials;

• measured train tracks;

• generalized interval exchange transformations;

• small actions of surface groups on metric trees.

There is yet another point of view on measured geodesic laminations [87] which is not discuss here.

12.1 Euclidean motivation

In this section we describe spaces of measured foliations and geodesic laminations on the 2torus S, how they compactify the Teichmüller space of S and how elements of the mapping class group of S act on this compactification. It is helpful to keep in mind this description since it is the simplest version of what we will do in the case of hyperbolic surfaces.

Consider a flat torus S of unit area. We will identify the space of all (nonoriented) closed geodesics C(S) with the set of primitive elements of (H_{1}(S, ℤ))/±1 (where zero corresponds to the empty geodesic). Define the geometric intersection number i(α, β) between elements of C(S) to be the number of points of transversal intersection. This is the same as the minimal number of points of intersection between loops α′, β′ in the homotopy classes of α, β. We have the intersection form Q

Q : H_{1}(S, ℝ) × H_{1}(S, ℝ) → ℝ

The absolute value of the restriction of this form to C(S) gives the geometric intersection number between α and β:

|Q(α, β)| = i(α, β)

Let α, β ∈ C(S) be a pair of closed geodesics with the intersection number 1. Then any element γ ∈ H_{1}(S, ℤ) is uniquely determined by the pair

(Q(γ, α), Q(γ, β))

However, elements γ of C(S) are not uniquely determined by

f(γ) := (i(γ, α), i(γ, β))

The ambiguity is: f(nα+mβ) = f(−nα+mβ) (here we treat elements of C(S) as homology classes). Thus there are at most two distinct elements of C(S) in f^{−1}(z, w). To avoid this ambiguity we take into account intersection with the third geodesic δ corresponding to the homology class α + β. Then

Q(nα + mβ, α + β) = n + m, Q(−nα + mβ, α + β) = −n + m

and i(nα + mβ, δ) = i(−nα + mβ, δ) iff either n = 0 or m = 0, which means that the triple

(i(γ, α), i(γ, β), i(γ, δ))

uniquely determines γ ∈ C(S).

The projectivized homology group P H_{1}(S, ℝ) is homeomorphic to the unit circle and the set:

{a/|a| : a ∈ C(S)}

is dense in P H_{1}(S, ℝ).

Choose α, β as generators of π_{1}(S). The word norm |x| (i.e. the distance between 0 and x ∈ π_{1}(S) with respect to the word metric) can be calculated geometrically as sup(i(x, α), i(x, β)). The elements of the space H_{1}(S, ℝ) can be represented as measured oriented geodesic foliations. Here the word “measure” means “a transversal measure”. For example, if a represents a simple closed geodesic of the length |a|, then it determines a foliation of S by geodesics parallel to a. Suppose that J is any finite geodesic arc in S transversal to a. Then we define mes_{a}(J) as Length(J) · |a|. To define the transversal measure on smooth arcs transversal to a we assume that the transversal measure does not change if we slide the end–points of J along the foliation and keep J transversal to the foliation a. The space of projective classes of measured geodesic laminations on S is the same thing as the space P H_{1}(S, ℝ) of (nonoriented) classes of geodesic foliations on S. I recall that the Seichmüller space T (S) of the torus S is naturally identified with the hyperbolic plane. Assuming that S is the origin in T (S) we conclude that P H_{1}(S, ℝ) serves as a natural compactification of T (S). Namely, suppose that we are given a direction x ∈ P H_{1}(S, ℝ). Think about x as a direction in the universal cover ℝ^{2} of S. Then we renormalize the metric on ℝ^{2} keeping the same action of the fundamental group so that we squeeze the metric in the direction x and stretch it in the orthogonal direction (to keep the area equal to 1). This sequence of metrics on the torus S has a limit: a degenerate bilinear form on ℝ^{2} which is defined up to a multiple. The only information about this form is that it vanishes on tangent lines which are parallel to x. It corresponds of projective class of a measured geodesic foliation on S.

Now we consider the classification of elements of infinite order in the classical modular group SL(2, ℤ) (recall that P SL(2, ℤ) is the mapping class group of the torus). Each hyperbolic element A ∈ SL(2, ℤ) has 2 fixed points in P H_{1}(S, ℝ): one repulsive [A_{−}], and another attractive [A_{+}]. Both are represented by eigenvectors of A. These eigenvectors give rise to a pair of irrational foliations on the torus S. We will identify H_{1}(S, ℝ) with the universal cover ℝ^{2} of the torus S so that A is a linear transformation.

Given the diffeomorphism A, the flat metric on the torus S can be chosen so that A is symmetric and thus repulsive and attractive directions are orthogonal. Stretching–squeezing the metric in these directions corresponds to moving along a geodesic in T (S). Suppose that x ∈ H_{1}(S, ℤ). Then we can apply to x iterations A^{n}(x). The renormalized sequence of vectors x_{n} = A^{n}(x)/|A^{n}(x)| is convergent to the direction [A_{+}]. The renormalized sequence y_{n} = A^{−n}(x)/|A^{−n}(x)| is convergent to [A_{−}]. We choose representatives A_{+}, A_{−} in the projective classes [A_{+}], [A_{−}]. Then A(A_{+}) = λ_{+}A_{+} and A(A_{−}) = λ_{−}A_{−}, where λ_{±} are the eigenvalues of A. The foliations A_{+} and A_{−} are “binding” the surface S: any nonzero foliation has nonzero intersection number either with A_{+} or with A_{−} (the corresponding eigenvectors are linearly independent).

Diffeomorphisms of the torus S which possess such pair of attractive/repulsive foliations are called Anosov.

In contrast, if A is a parabolic element, then it has only one fixed point; therefore the corresponding foliation does not bind S. The element A is an iteration of Dehn twists along a simple loop a, the foliation A_{±} has closed leaves which represent the homotopy class of a.

The goal of the rest of this Chapter is to extend this picture to the case of hyperbolic surfaces. Such extension was implicit in earlier works of Dehn and Nielsen (see [45]). In the modern time it was discovered and brought back to mathematics by Thurston.

12.2 Geodesic currents

Let M_{∞} denote the space of unordered pairs of distinct points on 𝕊^{1}:

M_{∞} := {(z, w) ∈ 𝕊^{1} × 𝕊^{1} : z ≠ w}/(z, w) ∼ (w, z)

Topologically M_{∞} is just the Moebius band. We identify 𝕊^{1} with the ideal boundary of the hyperbolic plane ℍ^{2}. Then the group P SL(2, ℝ) acts on M_{∞}. Recall that a Radon measure on a space X is a measure which defines a continuous functional on the space of continuous functions with compact support (with the topology of uniform convergence). Fix a discrete torsion-free group G ⊂ P SL(2, ℝ) such that ℍ^{2}/G = S is a hyperbolic surface. A geodesic current μ on S is a G-invariant Radon measure on M_{∞}. Let GC(S) denote the space of all geodesic currents. Clearly, sums of geodesic currents and their positive multiples again belong to GC(S). The space GC(S) of geodesic currents has a natural week^{∗} topology of convergence on continuous functions. More on geodesic currents see in [24], [25].

At this point the definition of geodesic current may look abstract and useless, however later on we shall see how it corresponds to more geometric definitions of measured laminations and foliations.

The Basic Example:

Suppose that g ∈ G is a primitive nontrivial element representing a simple closed geodesic γ on S (if S has boundary we assume that γ is non-peripheral). Take the G-orbit of the fixed-point set F ix(g) ∈ M_{∞}. This orbit forms a discrete subset E in M_{∞} and we can choose the Dirac measure on this orbit. This is the simplest example of a geodesic current on S (which is the sum of δ-functions supported on the points of E).

Consider the square M_{∞}^{2} := M_{∞} × M_{∞}, elements of this space correspond to pairs of geodesics in ℍ^{2}. In the space M_{∞}^{2} take the open subset IM_{∞}^{2} corresponding to pairs of geodesics which have transversal intersections in ℍ^{2}. The group G acts naturally on IM_{∞}^{2} and it is easy to see that this action is properly discontinuous. Suppose now that μ, ν are geodesic currents in GC(S), the product of these measures defines an G-invariant measure μ × ν on the open subset IM_{∞}^{2} of M_{∞}^{2}. Finally we project this product measure to IM_{∞}^{2}/G and take the total mass of the quotient measure. The result is called the intersection number i(μ, ν) between the currents μ, ν.

Theorem 12.1 (F. Bonahon, [24], [25]). The function

i : GC(S) × GC(S) → ℝ_{+}

is continuous and bilinear.

Exercise 12.2 If μ, ν are geodesic currents corresponding to distinct simple closed geodesics α and β (as in the Basic Example) then i(μ, ν) equals the number of points of intersection between α and β.

12.3 Measured foliations on hyperbolic surfaces

Let S be a compact hyperbolic surface, possibly with nonempty geodesic boundary. (The discussion in the case of surfaces of finite area is similar, see [73].) Roughly speaking, a measured foliation on S is a singular C^{1}–foliation with singularities similar to singularities of holomorphic quadratic differentials which is given a transversal measure. More precisely, suppose that P = {p_{1}, .., p_{m}} is a finite subset of S, which is the singular set of the foliation. If S has nonempty boundary we require each boundary circle to contain at least one point of P. Pick a coordinate covering U_{j} on S − P and assume that in each U_{j} we are given a real–valed C^{1}–function v_{j} such that |dv_{j}| = |dv_{i}| on U_{j} ∩ U_{i}. Assume also that for each singular point p_{s} there is a coordinate neighborhood V with complex coordinates such that |dv| = |Im(z^{k/2}dz)| for some positive integer k. Leaves of the foliations are the graphs immersed S in along which the functions v_{j} are constant (i.e. they are defined the same way we defined trajectories of quadratic differentials). We require each boundary circle of S to be contained in a singular leaf. We define lines in S and the universal cover S̃ of S the same way we defined them for quadratic differentials (Definition 5.3): they are either nonsingular trajectories or some particular subsets of singular trajectories which are images of ℝ under local isometry.

Exercise 12.3 Let L ⊂ S̃ be a lift of a boundary loop of S to the universal cover S̃. Then L is not a line.

Then |dv| is called a measured foliation. If J is an arc in S then the transversal measure (height) h_{v}(J) of J with respect to the foliation |dv|, is the integral

∫_{J} |dv|

The height h_{γ}(|dv|) of a (free homotopy class) of a loop γ on S is the infimum of heights of loops in [γ], this defines a map

h : (γ, |dv|) ↦ ℝ

Alternatively one can describe a measured foliation as follows. Let λ be a foliation on S (which is smooth away from P) with the set of isolated singularities P. We require that near each point p_{j} ∈ P the foliation λ looks like the image of the vertical foliation of ℂ under the map z ↦ z^{2/k}, the number k is called the degree of the singular point p_{j} := 0 ∈ ℂ. See Figure 24. We require each boundary circle of S to be contained in a singular leaf of λ.

Figure 24: Measured foliation near the critical point p_{i} where k = 3.

Now suppose that S − P is covered by charts so that images of leaves of the foliation are vertical lines in ℝ^{2} = ℂ and transition maps not only preserve the vertical directions but also are isometries in the horizontal direction (may be orientation reversing). Then the measure of an arc J contained in the coordinate neighborhood is the Lebesgue measure of its projection to the horizontal direction. The measure is invariant under sliding end-points of J along leaves of the foliation. This gives λ a transversal measure.

Definition 12.4 Critical leaves of the foliation λ are those which contain at least one singular point p_{j} ∈ P.

Basic examples of measured foliations appear from holomorphic quadratic differentials ϕ(z)dz^{2} on the surface S (if S is a Riemann surface without boundary) as follows:

|dv| = |Re(\sqrt{ϕ(z)}dz)|

This defines a map q : ϕ ↦ |dv|. Leaves of the foliation |dv| are vertical trajectories of this quadratic differential, i.e. the lines on which the quadratic differential has zero real part and positive imaginary part. Things are more complicated if we have a compact orientable surface S̄ with boundary. Let S be the interior of S̄, introduce a complex structure of finite type on S. Let ϕ be a holomorphic quadratic differential on S which has at worst simple poles at the punctures. Then as above we define measured foliation F on the surface S (which is the horizontal foliation of ϕ). Let Ŝ denote the conformal compactification Ŝ of S.

Figure 25: Blow up of a puncture.

Let p_{j} be a puncture of S (we identify p_{j} with zero in ℂ) where ϕ(z) = z^{k_{j}}ψ(z), k_{j} ≥ −1, ψ(z) is holomorphic at p_{j} and ψ(p_{j}) ≠ 0. Then we blow up each puncture p_{j} ∈ Ŝ into a circle γ_{j}. The result is diffeomorphic to the surface S̄, the foliation F extends to S̄ where we treat each circle γ_{j} as a part of a leaf. Since ϕ has at worst simple poles at the punctures, each γ_{j} contains at k_{j} + 2 singular points. In the Figure 25 we describe the blow up for k_{j} = −1, 0.

Figure 26: Whitehead move.

Two measured foliations |du|, |dv| are called measure–equivalent if heights of (free homotopy classes) of all simple closed curves on S with respect to these foliations are equal, i.e.

h(γ, |du|) = h(γ, |dv|), for each loop γ in S

Equivalently one can use the Whitehead equivalence relation on singular foliations by collapsing compact critical intervals to points (Whitehead move) and taking isotopy of foliations.

(See [93], [73], [59] for details.) It is clear that Whitehead moves preserve measure equivalence classes, the opposite direction is not obvious. In Figure 27 we give an example (taken from [93]) of a measured foliation which is not isotopic to the foliation of a holomorphic quadratic differential but is Whitehead equivalent to it.

Figure 27: Example of a measured foliations L on the surface S of genus 2 which is not isotopic to the horizontal foliation of a quadratic differential. Complement of the graph T on the surface S is the disjoint union of two annuli A_{1}, A_{2}. These annuli are foliated by horizontal loops. The transversal measure is given by the “height” of the transversal interval. Existence of a Euclidean metric with conical singularities corresponding to this foliation would force the equality l_{2} = l_{3}. Collapsing the intervals l_{2}, l_{3} to points makes L the horizontal foliation of a quadratic differential.

Let MF(S) denote the space of all measure equivalence classes of measured foliations on the Riemann surface S. For convenience we include in MF(S) the empty foliation (with zero transversal measure). We have a collection of height functions

h_{γ} = h(γ, ·) : MF(S) → ℝ_{+}

The topology on MF(S) is the topology of pointwise convergence of functions h_{γ} on C(S), which is the space of homotopy classes of simple loops on S.

Theorem 12.5 below implies that there is a finite collection of simple closed loops γ_{1}, ...., γ_{k} on S so that it is enough to verify convergence for the finite set of functions h_{γ_{j}}, 1 ≤ j ≤ k.

Theorem 12.5 (See [59], Expose VI.) There exists a finite collection of simple loops Γ := {γ_{1}, ..., γ_{k}} on S so that the height function h : C(S) × MF(S) → ℝ is determined by its restriction to Γ. The space MF(S) is homeomorphic to ℝ^{−3χ(S)−n}, where χ(S) is the Euler characteristic of S and n is the number of boundary circles.

Suppose now that S is oriented, we give S a complex structure. Let Q(S) denote the space of all holomorphic quadratic differentials on S whose nonsingular trajectories are disjoint from the boundary of S.

Theorem 12.6 (J. Hubbard and H. Masur) If the surface S is oriented then any measured foliation is equivalent to the foliation of vertical trajectories of a quadratic differential. Moreover the natural map

q : Q(S) → MF(S)

is a homeomorphism.

Proof of this theorem see in [93] and [73].

Recall that lines of the quadratic differential ϕ lift to quasi-geodesics of the hyperbolic metric in ℍ^{2} (see Section 5.3). In particular, each line in the universal cover S̃ of S has well-defined end-points on ∂_{∞}ℍ^{2} = 𝕊^{1}.

Now, given a measured foliation F on S, we will construct a geodesic current on S. In the case of orientable surface S we will assume that F is the vertical foliation of a holomorphic quadratic differential and ρ is the associated flat metric with conical singularities, otherwise we first pass to an orientable finite cover of S.

Suppose that F̃ is the lift of the foliation F to the hyperbolic plane. Let E = E(F) denote the set of unordered pairs (z, w) ∈ M_{∞} such that z, w are the end-points of a line ⟨z, w⟩ of F̃ in ∂ℍ^{2}. (This line may be critical and it could be that several lines have the same end-points). Let’s see that the set E is closed in M_{∞}. Suppose that ⟨z_{n}, w_{n}⟩ is a sequence of lines connecting points (z_{n}, w_{n}) ∈ E so that

(z, w) = lim_{n→∞}(z_{n}, w_{n})

Then compactness of the ideal boundary ∂_{∞}ℍ^{2} implies that a subsequence in ⟨z_{n}, w_{n}⟩ is convergent to a ρ-geodesic ⟨z, w⟩ which must connect the points z, w. This convergence is Hausdorff on compacts, therefore ⟨z, w⟩ must be a line. This proves that E is closed. Note that if L ⊂ S̃ is a leaf of boundary geodesic of S then L is not a line.

Now we construct a measure on E. For a subset B ⊂ M_{∞} we shall denote by L(B) the union of all leaves of F̃ which connect points of B. Suppose that K is a compact subset in M_{∞} which corresponds to the product of two disjoint intervals in 𝕊^{1}. Then there exists a compact K′ ⊂ ℍ^{2} such that all leaves of F̃ connecting points of K intersect K′. Leaves of F̃ are quasigeodesics in the hyperbolic plane, therefore there exists a compact “product” neighborhood C of (z, w) ∈ K such that L(C) ∩ K′ does not contain any critical points. Thus all these leaves in K′ intersect orthogonally an interval J ⊂ K′ and the intersection L(K)∩J is closed. We declare μ_{∞}(C) to be μ(J ∩L(C)) where μ is the transversal measure to F. Thus we have defined the measure on compact subsets of M_{∞} which are covered by products of small arcs in 𝕊^{1}. There is a standard procedure of extension of this measure to a Borel measure on M_{∞} which is essentially the same as the construction of the Lebesgue measure on ℝ^{n} starting with rectangles (see for instance [119]). The resulting measure μ_{∞} is supported on E and is G-invariant.

The set E also has the following important property. Suppose that (x, y), (z, w) ∈ E, then the hyperbolic geodesics γ_{x,y} and γ_{z,w} connecting x ot y and z to w do not intersect.

A measured foliation F on the surface S is called rational if it has only finitely many finite leaves connecting singular points and all other leaves are closed. (See Figure 28.) Clearly this property is invariant under Whitehead moves. In the case of tori this class of foliations consists of rational foliations (where all leaves are closed).

Theorem 12.7 (See [59].) Rational foliations are dense in MF(S).

It is easy to see that rational foliations on S are exactly those for which E(F) is a discrete set.

Figure 28: Example of a rational foliation on the surface of genus 2: double this foliation on the pair of pants.

12.4 Train tracks

Let S be a compact hyperbolic surface with (possibly empty) geodesic boundary, T is a finite trivalent graph. The image τ of an embedding T → int(S) is called a train track in S if the following conditions are satisfied:

1. τ is C^{1}-smooth outside of its vertices.

2. Near each vertex the graph τ is C^{1}-equivalent to

{(x, 0), −∞ < x ≤ 0} ∪ {(x, x^{2}), x ≥ 0} ∪ {(x, −x^{2}), x ≥ 0}  (38)

(see Figure 29).

3. Suppose that C is a component of cl(S −τ), D(C) is the double of C along C^{1}-smooth boundary edges of C (we exclude cusp points). The double D(C) has negative Euler characteristic.

Figure 29: Local picture of a train track.

Vertices v_{i} of τ are called switches and edges b_{j} are called branches. Local branches near v_{i} are components of τ − v_{i} in the local model (38). Note that for each switch there are 3 local adjacent branches, two of them are “outgoing” (they correspond to quadratic components in (38)) and one is “incoming” (it corresponds to the linear component in (38)).

A transversal measure μ for τ is an assignment of nonnegative real numbers μ_{j} to branches b_{j} of τ. These numbers must satisfy the following “conservation law” in each vertex v of τ:

if b_{1}, b_{2} are outgoing local branches at v and b_{3} is the incoming branch then:

μ_{1} + μ_{2} = μ_{3}

The support of this measure is a subgraph of τ consisting of branches with nonzero transversal measure. The pair (τ, μ) is called a measured train track. We will consider train tracks in S up to C^{1}-isotopy. More general equivalence relation involves transformation similar to the Whitehead move is reqired to establish equivalence between the categories of measured train tracks and measured laminations, we do not discuss it here. For a comprehensive discussion of measured train tracks see [166].

12.5 Measured geodesic laminations

Our discussion of geodesic laminations in this section mainly follows [38]. Let S be a compact hyperbolic surface possibly with geodesic boundary, let P T (S) denote the projectivized tangent bundle of S. A geodesic lamination λ on S is a compact subset of int(S) which is a union of complete disjoint geodesics which are called the leaves of λ.

Remark 12.8 The reason to require the lamination to be disjoint from the boundary of S is to make the definition consistent with the definition of a measured foliation where we assume that each boundary loop is a singular leaf.

Example 12.9 A simple closed geodesic in int(S) is a geodesic lamination.

Definition 12.10 A geodesic lamination λ is called maximal if it a maximal geodesic lamination with respect to inclusion.

Consider the space GL(S) of all geodesic laminations on S. Since geodesic laminations are closed subsets of the metric space S, we use Hausdorff distance to define to topology on GL(S). If S is closed then GL(S) is clearly compact since S is compact. In the case of surfaces with boundary is slightly more complicated since we assume that any geodesic lamination on S is disjoint from the boundary of S.

Exercise 12.11 Let γ be a boundary loop of S. Show that there is a constant ϵ > such that if a geodesic α ⊂ S contains a point x which is within the distance ≤ ϵ from γ then

• Either α = γ;

• or α crosses itself;

• or some lifts of α, γ are asymptotic to each other.

Thus there exists a compact subsurface S_{c} ⊂ int(S) so that any geodesic lamination in S is contained in S_{c}.

Exercise 12.12 Each geodesic lamination is contained in a maximal geodesic lamination.

Exercise 12.13 Suppose that λ is a maximal geodesic lamination on S. Then the closure of each complementary region C in S − λ is either isometric to the ideal triangle in the hyperbolic plane or (in case C contains aboundary geodesic γ of S) the region C is isometric to the “corona”, see Figure 30. In the latter case the interior of C is homeomorphic to the open annulus and has two boundary geodesics: γ, α so that the opposite geodesic rays in α are asymptotic to each other.

Figure 30: The corona.

Boundary leaves of λ are those leaves which belong to the boundary of one of the complementary regions S − λ. (Recall that the boundary geodesics of S do not belong to λ.) Each point x ∈ λ belongs to a single leaf λ_{x} of λ. Therefore we have a well-defined map σ : λ → P T (S). This map associates with the point x ∈ λ the direction tangent to λ_{x} at x. Using the fact that all geodesics in λ are disjoint, it is easy to conclude that σ is a continuous map.

Lemma 12.14 Any geodesic lamination λ is a proper subset of S.

Proof: The assetion is clear in the case of surfaces with boundary. Thus we assume that S is closed and λ is a geodesic lamination on S which covers the full surface S. Then the bundle P T (S) admits a globally defined continuous section σ. However the double cover of P T (S) is the unit tangent bundle to S. Therefore the Euler number e(P T (S)) = 2χ(S) ≠ 0. Contradiction. □

Lemma 12.15 Any geodesic lamination L on a compact hyperbolic surface S has empty interior.

Proof: Let L̃ denote the lift of L to ℍ^{2}. Choose an orientation for leaves of L̃ which varies continuously from leaf to leaf. Fix a geodesic segment J in ℍ^{2}, then the angles of intersection between leaves of L̃ and J vary continuously. We assume that geodesics in L̃ are parameterized by the arc-length. Then for each point x ∈ J we have a well-defined map ϵ(•, t) : x ↦ γ_{x}(t) where γ_{x} is the geodesic leaf of L̃ intersecting J at x. This map exponentially expands the distance and it is continuous since the section σ is continuous.

If J is entirely contained in L̃ and is transversal to one of the geodesics in L̃, then we apply to each point x ∈ J the map ϵ(x, t). For large t the image of ϵ covers a fundamental domain of F = π_{1}(S), which means that L covers the whole surface S. Contradiction. □

A geodesic lamination L is called minimal if every ray of each leaf of L is dense in L.

Lemma 12.16 : Properties of geodesic laminations. Suppose that L is a geodesic lamination . Then:

1. The number of boundary leaves of L is finite.

2. The Lebesgue measure of L is equal to zero.

3. The intersection of L with any transversal geodesic segment is a totally disconnected subset of zero measure.

4. The union of all boundary leaves is dense in L.

Proof: We prove this lemma in case of closed surfaces, the general case is proven by doubling the surface S along the boundary.

(1) The complement to L in S has finite hyperbolic area, thus it is the union of a finite number of hyperbolic surfaces with boundary which have finite area. They can have only finite number of boundary geodesics.

(2) Triangulate each complementary surface in S − L using ideal triangles only. We get a new lamination L′ which contains L. Extend L′ in each ideal triangle to a singular foliation F with a single singular point inside of each triangle, see Figure 31.

Figure 31: The extension of measured lamination.

The winding number of F around each singular point is equal to ±1. Hence −2χ(S) = |e(P T (S))| ≤ n, where n is the number of singular points which is the same as the number of ideal triangles. The area of each ideal triangle is π. Therefore the area of S −L is at least nπ ≥ −2πχ(S). However we know that Area(S) = 2|χ(S)|; this implies that n = −2χ(S) and Area(S) = Area(S − L). Thus Area(L) = 0.

The property (3) follows from (2). In any Cantor subset C ⊂ [0, 1] the set of boundary points of complementary intervals is dense, it implies (4). □

One can visualize neighborhoods of a geodesic lamination λ using “fat train tracks”. Namely, if λ is a geodesic lamination and ϵ > 0 is sufficiently small, then the ϵ-neighbor-hood Nbd_{ϵ}(λ) is a “fat train track” around λ. Note that for a fixed ϵ we can find a very long simple closed geodesic λ on S, so that the ϵ-fat train-track around λ has large (nonabelian) fundamental group. It is most convenient to see the structure of a geodesic lamination using the fundamental domain D of the group F. Let λ̃ denote the lift of λ to the hyperbolic plane. Then λ̃ ∩ D is a disjoint union of geodesic segments. The ϵ-fat train track is the union of “rectangles” (“flow boxes”). Inside of each rectangle the lamination can be completed to a vertical foliation (see Figure 32). If ϵ is sufficiently small then on the top (and bottom) of each flow box we have not more than two other flow boxes attached. The corresponding segment (top or bottom) is called a switch, if this number is exactly two. Suppose for concreteness than we have two rectangles R_{1}, R_{2} attached to the top segment σ of the rectangle R_{3}. At the switch σ the rectangle R_{3} is (locally) incoming and the rectangles R_{1}, R_{2} are (locally) outgoing. To get the usual train tracks from fat ones we collapse each flow box along horizontal segments. The result is a finite graph in S. To get a train track structure for this graph we assign the “incoming/outgoing” labels to local branches emanating from vertices according to the labels on the original fat train track.

Figure 32: Train tracks in the fundamental domain. Each track is the union of rectangles (“flow boxes”). Inside of each rectangle the lamination has especially simple “product” structure, where all tracks (leaves of the lamination) are almost parallel. Switch is the place where tracks in different boxes are starting to diverge. With respect to the product structure inside of each flow box, the transversal measure corresponds to a measure supported on a Cantor set which is the projection of the lamination on the bottom of the rectangle.

Note that compactness of GL(S) implies that there is a finite collection of “fat train tracks” T_{1}, ..., T_{k} on S so that each geodesic lamination is contained in one of these tracks.

A measured geodesic lamination on S is a geodesic lamination L together with a transversal measure μ supported on L. Here is one way to define it (see [166]). Given L we consider a collection J of all compact smooth 1-dimensional submanifolds in S with end-points in S − L which are transversal to L. A transversal measure μ is a function

μ : J → ℝ_{+}

which has the following properties:

• On each J ∈ J the restriction μ|_{J} is a σ-additive Borel measure;

• μ(J) = μ(J′) if J, J′ ∈ J are isotopic through elements of J;

• For each J ∈ J, μ(J) > 0 iff J ∩ L ≠ ∅.

Here is another way to think about transversal measures. Cover the lamination L by a finite union of flow boxes forming a fat train track τ̂. Inside of each flow box the lamination is the product of a Cantor set (or a finite set) and a vertical segment. The transversal measure corresponds to a Borel σ-additive measure whose support is this Cantor (or finite) set. Equivalently speaking, μ defines a Borel measure on any compact geodesic interval J ⊂ S which is transversal to λ. If J_{t} is a continuous family of such intervals so that end-points are contained in S − L, then the transversal measures on all J_{t} coincide.

After collapsing τ̂ to a train track τ ⊂ S, the transversal measure for L will produce a transversal measure for τ. (The conservation law follows from finite additivity of the transversal measure μ.) The advantage of train tracks (versus measured geodesic laminations ) is that they are purely combinatorial objects (unlike laminations which are usually transcendental objects).

The space of measured geodesic laminations ML(S) on the surface S has a natural week^{∗} topology given by evaluation of measures on various intervals in S. One can prove that this topology is equivalent to the topology defined via homotopy classes of simple loops on S similarly to the topology on MF(S). There is a natural map

♭ : ML(S) → GL(S)

The problem however is that ♭ fails to be both injective and surjective. Below we will see examples of geodesic laminations L which do not admit transversal measures supported on the whole L. A measured geodesic lamination (L, μ) = λ is called uniquely ergodic if any transversal measure for L is proportional to μ. The simplest example of a measured geodesic lamination which is not uniquely ergodic is one where L consists of two simple closed geodesics in S. Then the space of transversal measures for L can be naturally identified with

ℝ^{2} \ (ℝ × {0} ∪ {0} × ℝ)

There are also minimal measured geodesic laminations which are not uniquely ergodic. Similarly, a measured foliation is called uniquely ergodic if for the corresponding singular foliation F on S, any two transversal measures are proportional. H. Masur had proved in [136] that uniquely ergodic foliations form a set of full measure in MF(S).

Lemma 12.17 (1) Suppose that γ is a closed leaf of a measured geodesic lamination L on S. Then γ is isolated in L, i.e. for a small interval J transversal to γ, the intersection J ∩ L = J ∩ γ is a single point.

(2) If γ is not a closed leaf then γ is recurrent, i.e. it accumulates to itself in the both directions.

Proof: Suppose that γ is not isolated. Choose a lift γ̃ of γ to the universal cover ℍ^{2}. Let ⟨g⟩ denote the stabilizer of γ̃ in π_{1}(S). There is a sequence of leaves l_{n} of L̃ − γ̃ which is convergent to γ̃. First we suppose that end-points of all l_{n} are disjoint from the end-points of γ̃. If π_{γ} is the orthogonal projection to γ̃ then for large n the diameter of π_{γ}(l_{n}) is greater than the translational length ℓ(g) of the hyperbolic element g. It means that g(l_{n}) ∩ l_{n} ≠ ∅ (see Figure 33).

This contradicts the property that distinct leaves of L are disjoint. Now assume that for large n > n_{0} the end-point l_{n}^{+} of every leaf l_{n} is the repulsive fixed point of g. Let J be a small transversal interval to γ̃. Then we can assume that the interval J intersects L̃ by the orbit l_{0} = l, l_{1} = g(l), l_{2} = g^{2}(l), .... The restriction of the transversal measure on J is atomic and, by invariance, the mass of each point J ∩ L_{k} is the same nonzero number. This however contradicts the countable additivity of the transversal measure μ. □

This Lemma gives examples of geodesic laminations L which do not admit any transversal measures supported on the whole L. Lemma 12.17 implies the following:

Lemma 12.18 (Decomposition lemma.) Each measured geodesic lamination L is a disjoint union of a finite number of minimal sublaminations L_{j}.

Minimal components of L which are closed geodesics are called periodic, other components are called recurrent.

The concepts of geodesic lamination and measured geodesic lamination generalize to “blown up” geodesic lamination and “blown up” measured geodesic lamination : replace “geodesics” in the definition of geodesic lamination by “complete curves of constant curvature which do not lift to metric circles in ℍ^{2}”.

12.6 Train tracks and convergence of measured laminations

Suppose that (L, μ) ∈ ML(S) and τ̂ is a fat train-track around L, τ̂ = ∪_{j=1}^{n}B_{j} is the union of flow boxes. Then for each box B_{j} and horizontal interval J_{j} ⊂ B_{j} (transversal to L) we have a number μ^{j} = μ(J_{j}). If (L_{s}, μ_{s}) ∈ ML(S) is convergent to (L, μ) then for large n the leaves of L_{s} ∩ B_{j} will be transversal to all J_{j}. Thus we will have:

lim_{s→∞} μ_{s}(J_{j} ∩ L_{s}) = μ^{j}  (39)

One can prove the opposite as well. Namely, if (L_{s}, μ_{s}) is Hausdorff convergent to L and the equality (39) above holds, then (L_{s}, μ_{s}) → (L, μ).

Definition 12.19 A pair of measured geodesic laminations α, β is binding the surface S if

i(γ, α) + i(γ, β) > 0

for any γ ∈ ML(S) − {0}.

A pair of simple closed geodesics α, β is binding S if and only if S − (α ∪ β) consists only of simply-connected components and annular neighborhoods of boundary loops.

Take a binding pair of measured geodesic laminations α, β. We will consider the sum γ = α + β of the corresponding geodesic currents. Then γ is binding the surface S in the sense that if λ is a geodesic current on S then i(λ, γ) > 0.

Lemma 12.20 (F. Bonahon,[25]) If γ is binding then the set

C = {λ ∈ ML(S) : i(λ, γ) ≤ 1}

is compact.

Proof: The set C is clearly closed, so, by the definition of the weak^{∗} topology, it suffices to show that for every compact subset K ⊂ M_{∞} the set {λ(K), λ ∈ C} is bounded in ℝ. Since the set K is compact and γ is binding, we can cover K by a finite number of open neighborhoods U_{g} of geodesics g ∈ K so that for each g there exists a small open neighborhood U_{h} of a geodesic h contained in the support of γ such that:

• Each pair of geodesics g′ ∈ U_{g}, h′ ∈ U_{h} have nonempty transversal intersection.

• γ(U_{h}) > 0.

• U_{g} ∩ U_{h} project injectively to M_{∞}^{2}/G.

Thus 1 ≥ i(λ, γ) ≥ γ(U_{h}) · λ(U_{g}), which implies that

λ(U_{g}) ≤ [γ(U_{h})]^{−1}

□

12.7 From measured foliations to measured laminations

Two concepts: “measured foliations” and “measured laminations” are essentially equivalent. Namely, there exists a natural homeomorphism between the spaces MF(S) and ML(S). Suppose that F is a measured foliation on S with the lift F̃ to the hyperbolic plane. Recall that in the section 12.3 we have defined a closed subset E = E(F) ⊂ M_{∞} ( which is the set of end-points of leaves of F̃) and a G-invariant measure (geodesic current) μ_{∞} supported on E.

Now connect all points (z, w) in E = E(F) by hyperbolic geodesics γ_{z,w}. The result is a G-invariant closed subset L̃ in ℍ^{2}. No distinct geodesics in this subset intersect and if γ is a lift of a boundary geodesic of S then γ is disjoint from the closure of L̃.

Therefore L̃ is the lift to the universal cover S̃ of a geodesic lamination L in S. It remains to construct is a transversal measure λ for L. For this we push the measure μ_{∞} back to the hyperbolic plane. Namely, let J be any compact geodesic interval in ℍ^{2} which is transversal to L̃. Consider the union of all leaves of L̃ which intersect J. This union corresponds to a closed subset ϕ(J ∩ L̃) ⊂ E(F) ⊂ M_{∞}. Moreover, there exists a homeomorphism ϕ : J ∩ L̃ → ϕ(J ∩ L̃). Then define the transversal measure λ on J to be the pull back of μ_{∞} via ϕ from E(F). It is easy to see that this defines a transversal measure for L.

12.8 From laminations to foliations

Now go back from laminations to foliations. Let us first consider the case when λ = (L, μ) is a measured lamination on S such that L is a maximal lamination. Take a fat train-track neighborhood U of λ = (L, μ). Inside of U the lamination λ has an obvious extension to a measured foliation F′, namely each flow box U_{j} = I_{j}×I has a coordinate system: λ = C_{j}×I, where C_{j} is a Cantor set. Use a monotone continuous “Cantor” function f_{j} : C_{j} → I_{j} which is a homeomorphism everywhere except at boundary points of complementary intervals. We get a collection of continuous mappings (f_{j}, id) : C_{j} × I → U_{j} which define a continuous map from L to a foliation of U by “vertical segments” {t} × I ⊂ U_{j}.

Take any complementary triangle T ⊂ S − τ̂. Choose 3 points A, B, C on different sides of T which are not vertices. Then we collapse the triangle T in S to a tripod so that A, B, C are collapsed to a single vertex of the tripod. (See Figure 34.)

The result has a natural structure of a singular foliation F. We also get a natural map L → F, which is a homeomorphism everywhere except on boundary leaves. Hence we can push-forward the transversal measure μ from λ to F and get a measured foliation .

What to do in the case when L is not a maximal lamination? First of all add disjoint simple closed geodesics γ_{i} ⊂ int(S) to L so that each component of S′ = S − (∂S ∪ L ∪ γ_{i}) is either diffeomorphic to a disk or annulus or pair of pants. If C is a component of S′ which is bounded by 3 closed geodesics we collapse C to the “θ-graph” θ embedded in C so that C − θ is the disjoint union of three annuli. Then collapse C onto θ (see Figure 35).

Figure 34: Collapse of complementary triangles.

Figure 35: Collapse of complementary coronas.

In the case C has at least one ideal boundary vertex we add to C disjoint simple geodesics (asymptotic to the ideal vertices of C) to triangulate C into union of ideal triangles and coronas. (Some of these coronas may be adjacent to boundary loops of S.) Collapse each ideal triangle on ideal tripod as in the case of maximal laminations.

Finally we collapse each corona onto union boundary circle and of geodesic ray asymptotic to its vertex, see Figure 36.

Figure 36: Collapse of complementary coronas.

Now we split S open along the isolated closed geodesics γ in L and insert foliated annuli U(γ) with the Lebesgue transversal measure so that the total mass of this measure is μ(J), where J is a small transversal segment to γ. Now consider a complementary region C of λ′.

The map (L, μ) → F → E(F) ∈ GC(S) defines a homeomorphic embedding of ML(S) to the space of geodesic currents GC(S).

12.9 Action of Mod_{S}

Recall that each element of the Teichmüller modular group Mod_{S} is represented by a diffeomorphism. Quasiconformal diffeomorphisms of S naturally act on MF(S) by pushforward of transversal measures. (This means that for any F ∈ MF(S) and h ∈ Diff(S) we let h(F) be the image of the foliation F and to calculate the transversal measure of J with respect to h(F) we take the transversal measure of h^{−1}(J) with respect to F.)

Using geodesic currents one can describe this action as follows. Let E(F) be the geodesic current corresponding to F. Any quasiconformal homeomorphism h : S → S lifts to ℍ^{2} and then extends to a homeomorphism h_{∞} : 𝕊^{1} → 𝕊^{1}. Thus we get the “product” homeomorphism (h_{∞}, h_{∞}) : M_{∞} → M_{∞}. Then E(h(F)) = (h_{∞}, h_{∞})(E(F)) where (h_{∞}, h_{∞}) acts as push-forward on measures on 𝕊^{1} × 𝕊^{1}.

We conclude that the action does not depend on the homotopy class of h. Therefore we get a well-defined action of the modular group Mod_{S} on MF(S).

12.10 The geometric intersection number

Let C(S) denote the set of geodesic laminations on S which are finite unions of disjoint simple closed geodesics. We shall use the notation MC(S) for the collection of measured laminations which correspond to elements of C(S). To each L ∈ C(S) we assign the counting transversal measure, i.e. if J is an arc in S with end-points in S − L, then the transversal measure of J is the number of points of transversal intersection between J and L. Thus we get a canonical embedding C(S) ↪ MC(S).

Theorem 12.21 The union

MC(S) = ⋃_{γ∈C(S),r∈ℝ_{+}} rγ

is dense in ML(S).

Proof: See [55], [166]. □

Suppose that α, β are two simple closed geodesics on S. We define the geometric intersection number i(α, β) to be the number of points of transversal intersection between α and β. By homogeneity the function i extends to a function

i : MC(S) × MC(S) → ℤ

Theorem 12.22 (W. Thurston) There exists a continuous extension of i from MC(S) to ML(S). This extension is obtained by restricting the intersection number i from GC(S) to ML(S) (see Theorem 12.1).

Proof: See [24], [166]. □

This extension is also called the (geometric) intersection number and is denoted by i.

12.11 From laminations to trees

We first define blow up of a measured geodesic lamination λ = (L, μ).

Definition 12.23 For each isolated leaf γ of λ (which is a simple closed geodesic so that for a small interval J transversal to γ such that J ∩ L = J ∩ γ the tranverse measure μ(J) is positive) we do the following. Take a small tubular neighborhood U(γ) of γ so that U(γ) ∩ L = γ and foliate U(γ) by constant curvature curves equidistant from γ. We give the foliated annulus U(γ) the Lebesgue transversal measure so that the total mass of this measure is μ(J). The resulting measured lamination λ′ = (L′, μ′) of S (by curves of constant curvature) is called the “blow up” of λ.

Let L̃, L̃′ denote lifts of L, L′ to the universal cover ℍ^{2}. We introduce the following pseudo-metric d̃_{λ} on ℍ^{2}. Given any two points p, q we connect them by various piecewisegeodesic curves α(p, q) and minimize the transversal μ′-measure of α(p, q). Let’s see that this minimum is always realized. We construct a metric of nonpositive curvature on S̃ so that the complement to L̃′ is isometric to ℍ^{2} − L̃, the metric on foliated strips of L̃′ is flat and leaves of L̃′ are geodesics. Then geodesic segments of this metric give minimizing curves for the pseudo-distance d̃_{λ}.

Now we identify all points which are within zero distance. The quotient is a metric space T_{λ} with the metric d_{λ}. It is easy to see that we identify two points if and only if :

• either they are on the same leaf of L̃′,

• or in the same complementary component ℍ^{2} − L̃′.

The quotient topology coincides with the topology given by the metric d_{λ}. If J is an interval transversal to L̃′, then the projection J → T is given by a “Cantor” function J → T which is constant on all components of J − L̃′ and whose image is homeomorphic to J. It follows that each simple path in T can be lifted to an embedded piecewise-geodesic curve in ℍ^{2}. Therefore T is a metric tree: each triangle is a tripod (in degenerate case it could be a segment). The tree T = T_{λ} is called the dual tree to the lamination λ. The action of the group G = π_{1}(S) projects to an isometric action of G on the tree T. It is easy to see that this action is minimal: T is the union of axes of elements of G acting on T, the tree T is geodesically complete.

Lemma 12.24 The action of G on T is small.

Proof: Suppose that J is an interval in T which is fixed by a nontrivial subgroup G′ ⊂ G. Then for generic x ∈ J the preimage of x in ℍ^{2} is a single leaf of our foliation L̃′. This leaf must be invariant under G′. Therefore G′ is cyclic. □

As an example consider the following situation. Take a pair α, β of simple closed geodesics on S which we regard as measured laminations using the Dirac transversal measure oof unit total mass. Let g = g_{α} denote an element of π_{1}(S) corresponding to the loop α.

Exercise 12.25 Under the conditions above i(α, β) equals the translational length ℓ_{T_{β}}(g) of the element g acting on the tree T_{β}.

The following theorem was proven by R. Skora in [185], other proofs were given by J. P. Otal [162] and by B. Farb and M. Wolf [58]. Yet another proof is implicitely contained in the paper by F. Luo [126] who gave a characterization of translation lengths functions ℓ_{T} : G → ℝ for fundamental groups of hyperbolic surfaces which correspond to the trees T dual to measured laminations.

Theorem 12.26 Suppose that T is a small minimal π_{1}(S)-tree so that each peripheral element of π_{1}(S) acts as an elliptic isometry of T. Then there exists a compact “blown up” measured geodesic lamination λ ⊂ S so that its dual tree T_{λ} is equivariantly isometric to T.

We will present a proof of Skora’s theorem in Chapter 14 as an application of the Rips’ Theory (following [20]).

Corollary 12.27 There is a finite collection of elements γ_{1}, ..., γ_{k} of π_{1}(S) so that the equivariant isometry class of any small minimal π_{1}(S)-tree T (so that each peripheral is element of π_{1}(S) acts as an elliptic isometry of T) is uniquely determined by the translation lengths

(ℓ_{T} (γ_{1}), ..., ℓ_{T} (γ_{k}))

Let T (G) be the space of equivariant isometry classes of small minimal actions of G = π_{1}(S) on metric trees (so that each peripheral is element of G acts as an elliptic isometry). Each element T ∈ T (G) corresponds to a length function

ℓ_{T} : F → ℝ_{+}

We topologize T (F) using the product topology on the space of functions F^{ℝ} = Maps(F, ℝ). As we saw, it is enough to verify convergence on a finite number of elements of F. We conclude:

Theorem 12.28 The correspondence ML(S) ∋ λ ↦ T_{λ} ∈ T (F) is a homeomorphism.

12.12 Interval exchange transformations

In this section we describe yet another (dynamical) interpretation of measured foliations.

Let r = (r_{1}, r_{2}, ..., r_{n}) be a vector with positive coordinates and σ ∈ S_{n} be a permutation on n symbols. Let χ : {1, ..., n} → {−1, 1} be a map. We define generalized interval exchange transformation E_{r,σ,χ} associated with the data (r, σ, χ) as follows:

Let ρ = r_{1} + ... + r_{n}. Choose the points x_{0} = 0, ..., x_{i} = x_{i−1} + r_{i}, ..., x_{n} = x_{n−1} + r_{n} = ρ and the points y_{0} = 0, ..., y_{i} = y_{i−1} + r_{σ(i)}, …, y_{n} = y_{n−1} + r_{σ(n)} = ρ on the interval [0, ρ]. Note that the intervals [x_{i−1}, x_{i}] and [y_{σ^{−1}(i)−1}, y_{σ^{−1}(i)}] have the same length. Let

λ_{i} : [x_{i−1}, x_{i}] → [y_{σ^{−1}(i)−1}, y_{σ^{−1}(i)}]

be the isometry which is orientation-reserving if χ(i) = −1 and is orientation-preserving otherwise. Then E_{r,σ,χ} is the collection of maps λ_{i}, i = 1, ..., n. If χ(i) = 1 for each i then E_{r,σ,χ} is called the interval exchange transformation. The generalized interval exchange transformation is called irreducible if the permutation σ does not have an invariant subset {1, ..., k}, k < n.

You can find a detailed discussion of the interval exchange maps in relation to measured foliations in [186], [136], here I will give only a sketch.

Let S be a compact hyperbolic surface and (F, μ) a measured foliation on S. Choose a closed interval J ⊂ S transversal to F so that:

(a) If γ is an infinite ray in a leaf of F then γ ∩ J ≠ ∅.

(b) J does not contain singular points of F.

Existence of such interval J is clear in the case of minimal foliation F, the general case is a bit more complicated and we will not discuss it here.

Fix a coorientation on J; this defines locally a direction on trajectories of F emanating from the points of J. We call a point x ∈ J nonspecial if the “positive” trajectory of F emanating from x hits the interior of J in the positive direction before bumping into a singular point or an end-point of J. For each nonspecial point x we have a well-defined “first return” E(x) ∈ int(J), which is the point of the first intersection of the positive trajectory emanating from x with J. (See Figure 37.)

There are only finitely many singular points of F and critical trajectories leading to them from J; also there are only two end-points of J and only one direction leading to each of them. Suppose for example that two trajectories of F (starting on J) before the first return enter the same singular point from the same direction. Then one of these trajectories is contained in another one, therefore at least one of them has the 1-st return to J before entering this singular point.

Figure 37: The first return map E(x).

Thus the number of special points on J is finite. For each open maximal nonspecial interval J_{i} the map E is a Euclidean isometry E_{i} : J_{i} ↪ J (which must preserve the orientation if S is orientable). The map E defined outside of the set of special points of J determines a generalized interval exchange map. In Section 14.15 we will describe how to recover equivalence class of measured foliation from the corresponding interval exchange transformation.

12.13 An application of Skora’s theorem

Let S be a compact hyperbolic surface with geodesic boundary, G = π_{1}(S). Assume that we have an elliptic (relative to the peripheral subgroups of G) action G ↷ T of G on a tree T (i.e. each element of G corresponding to a peripheral loop on S acts as an elliptic isometry of T). Suppose also that T is a small minimal G-tree.

Theorem 12.29 Consider the collection C_{T} of all elements g ∈ G − {1} which have fixed points on T. Then (under the above conditions) there exists a subsurface B ⊂ S with incompressible boundary such that g ∈ C_{T} iff g is conjugate into π_{1}(B). The subsurface B is called the maximal elliptic subsurface in S.

Proof: According to Skora’s theorem, there exists a blown up measured geodesic lamination λ on S so that T = T_{λ}. Let λ̃ be the lift of λ to the universal cover ℍ^{2}. It is clear that for every hyperbolic element g ∈ G, whose axis A_{g} ⊂ ℍ^{2} intersects λ̃ transversally, the projection of A_{g} to T_{λ} is a geodesic on which g acts as a translation. Thus, g acts on T as an elliptic element if and only if one of the following is satisfied:

• either the axis A_{g} ⊂ ℍ^{2} of g is a leaf of λ̃;

• or A_{g} is contained in a component on ℍ^{2} − λ̃.

In each non-contractible component C_{j} of S − λ we take a compact subsurface B_{j} which is a deformation retract of C_{j}. The union of these subsurfaces is B. □

12.14 Characterization of aperiodic homeomorphisms

As we saw in the Section 12.1, each hyperbolic element α of the classical modular group SL(2, ℤ) can be represented by an Anosov diffeomorphism A of the torus T^{2}. The diffeomorphism A had two invariant foliations: one stable (attractive) and the other –unstable (repulsive). The goal of this section is to prove a similar statement for homeomorphism of hyperbolic surfaces.

Let S be a complete hyperbolic surface of finite area which has the genus g and m punctures. Take an aperiodic orientation-preserving homeomorphism h : S → S (see Section 2.9 for the definition). We will assume that S is orientable and h preserves orientation. Then h acts as an element h_{∗} of the modular group Mod_{S} on the Teichmüller space T (S).

Theorem 12.30 (L. Bers [18]) The isometry h_{∗} of the Teichmüller space has an invariant geodesic.

Proof: I give a proof following [18] and [1]. Let r : T (S) → M(S) denote the projection. Let γ denote the loop in the moduli space M(S) = T (S)/Mod_{S} which is covered by the geodesic segment [z_{0}, h_{∗}(z_{0})], where z_{0} is the origin of T (S). Let’s start deforming this loop in the given homotopy class trying to minimize the length. Formally speaking this is not a well-defined operation since the modular group does not act freely on the Teichmüller space. Therefore we just move a point z ∈ T (S) so that length of the geodesic segment [z, h_{∗}(z)] is decreasing. I assume that d_{T} (z_{0}, h_{∗}(z_{0})) = K. The Teichmüller space is complete. Therefore if the function d(z, h_{∗}z) has no minimum in the Teichmüller space, then there is a sequence of points z_{n} → ∂T (S) which is minimizing the distance d(z, h_{∗}z). Project this sequence of points and segments to the moduli space. Without loss of generality we may assume that the sequence r(z_{n}) ∈ M(S) is convergent to a point w on the Mumford’s compactification of the moduli space. Suppose that w corresponds to pinching of a collection of simple disjoint loops L on S. Then for sufficiently large n, the length of each of these loops on (S, z_{n}) is smaller than some number ϵ < 1/K.

If two geodesics on (S, z_{n}) have length < Kϵ then they are disjoint and there are at most 3g − 3 + m closed geodesics of the length < 1. Suppose that γ is a geodesic in L which has length at most 1/K(3g − 3). Then all iterates h^{k}(γ), 0 ≤ k ≤ 3g − 3 + m, have geodesic length at most 1. It follows that there are freely homotopic loops among them, therefore h is not aperiodic.

So, there is a point O ∈ T (S) so that d(O, h_{∗}(O)) is minimal. If h_{∗}(O) = O, then h acts as a conformal automorphism of O and hence h_{∗} : π_{1}(S) → π_{1}(S) has finite order. This contradicts the assumption that h is aperiodic. Otherwise, O ≠ h_{∗}(O) and the geodesic in T (S) through the points O, h_{∗}(O) is h-invariant. □

It turns out that the invariant geodesic is actually unique:

Theorem 12.31 (A. Marden, K. Strebel, [130]) Any aperiodic orientation-presering homeomorphism h : S → S has a unique invariant geodesic in T (S).

Consider the point O on the invariant geodesic L ⊂ T (S) as the origin S of the Teichmüller space T (S). Assume f : S → S is the extremal Teichmüller map homotopic to h. It follows that f^{2} is also a Teichmüller map and K(f^{2}) = K(f)^{2}. According to Teichmüller’s description of extremal quasiconformal maps (see Section 5.4), there exists a holomorphic quadratic differential ϕ on S such that the point (S, f) on L is given by the Beltrami differentials

μ = t\overline{ϕ}/|ϕ|, t ∈ ℝ

Local calculations imply that the Beltrami differential of f^{−1} is given by

−t\overline{ϕ}/|ϕ|, t ∈ ℝ

It follows that final and terminal quadratic differentials of the map f coincide (up to a positive multiple).

Recall what the above assertion means for the points O, h_{∗}(O). Let K be the coefficient of quasiconformality of f. Assume that the conformal structure on S is given by O. Then the quadratic differential ϕ defines a flat metric with conical singularities on S and a measured foliation F^{−} consisting of vertical trajectories of ϕ. The set of singular points of this foliation is invariant under the solution f of the Beltrami equation

\overline{∂}h/∂h = t\overline{ϕ}/|ϕ|

In the nonsingular part of the surface S the map h is affine with respect to the piecewise-Euclidean metric. It stretches the distance by k = \sqrt{K} > 1 in the horizontal direction and squeezes it by k^{−1} in the vertical direction. However the surface S has also a second measurable foliation F^{+} given by the horizontal trajectories of ϕ (or equivalently by vertical trajectories of the differential iϕ). These two foliations are transversal (in nonsingular points), have the same set of singular points and have no common leaves. Considering F^{±} as measured foliations we conclude that:

h(F^{+}) = kF^{+}, h(F^{−}) = k^{−1}F^{−}, where k > 1  (40)

(see Section 12.9 for description of the action of elements of Mod_{S} on measured foliations). If the surface S is not compact then the quadratic differential ϕ has at worst simple poles in the punctures of the surface S. Hence we can compactify the surface S by adding one circle for each puncture so that F extends to a measured foliation F̄ on the resulting compact bounded surface S̄.

Figure 38: Surface S of genus 2 is a 2-fold ramified covering over the torus with irrational foliation L. The singular foliation L̃ on S has 2 critical points, one of them is the center of the fundamental domain, the other one corresponds to the corner. The foliation L is the stable foliation of an Anosov diffeomorphism f of the torus. f lifts to a pseudo-Anosov homeomorphism of S with the stable foliation L̃.

Let S be a compact hyperebolic surface (possibly with geodesic boundary).

Definition 12.32 A homeomorphism h : S → S is called pseudo-Anosov if it has two projectively invariant measured foliations F^{+}, F^{−} such that the conditions (40) are satisfied, foliations F^{+}, F^{−} are “transversal” and h is a diffeomorphism outside of the singular set of these foliations. The foliation F^{+} is called stable and the foliation F^{−} is called unstable foliation of h. The number k is called the stretch factor of h.

Thus we have the following:

Theorem 12.33 (W. Thurston) Suppose that S is a compact oriented hyperbolic surface with geodesic boundary and h : S → S is an orientation-preserving aperiodic homeomorphism. Then h is pseudo-Anosov.

Remark 12.34 The assumptions that S is oriented and h is orientation-preserving are unnecessary, see [38], however we will not need this more general result.

Theorem 12.35 (See [59]) Stable and unstable foliations of any pseudo-Anosov homeomorphism are uniquely ergodic.

There are numerous examples of pseudo-Anosov homeomorphisms of surfaces, in a sense any generic element of Mod_{S} is aperiodic. The simplest example is given by branched coverings over the 2-torus S → T, where we lift an Anosov diffeomorphism A : T → T to a pseudo-Anosov homeomorphism of S, see Figure 38.

It is also true that any pseudo-Anosov homeomorphism is aperiodic, we will prove this below.

Theorem 12.36 Any pseudo-Anosov homeomorphism f : S → S of compact hyperbolic surface with (possibly empty) geodesic boundary, is aperiodic. The stable/unstable foliations F^{+}, F^{−} of f do not have closed trajectories.

Proof: Let F^{+}, F^{−} be the stable/unstable pair of foliations of f with the stretch factor k > 1. Note that for any nonperipheral loop γ on S,

h_{γ}(f_{∗}F^{+}) = kh_{γ}(F^{+}), h_{f_{∗}(γ)}(F^{+}) = kh_{γ}(F^{+})

where h_{γ}(•) is the hight function. Thus f_{∗}^{m}(γ) is never homotopic to γ unless m = 0 or h_{γ}(F^{+}) = 0. However if h_{γ}(F^{+}) = 0 then γ is homotopic to a loop γ′ in the foliation F. Let γ := γ′. There are two possible cases.

Case A. The trajectory γ is not critical. Then γ is transversal to the foliation F^{−}. There are only finitely many maximal foliated annuli in the foliation F^{+}, thus by taking if necessary sufficiently large power of f we get: f(γ) is a noncritical trajectory isotopic to γ through leaves of F^{+}. Then h_{f(γ)}(F^{−}) = h_{γ}(F^{−}) ≠ 0 since F^{+} (and in particular γ) is transversal to F^{−}. This is impossible.

Case B. The trajectory γ is critical. The foliation F^{+} has only finitely many closed critical trajectories and we repeat the same argument as above. □

12.15 Dynamics of pseudo-Anosov homeomorphisms

In this section we will discuss action of aperiodic elements of the Teichmüller modular group on the space of measured geodesic laminations ML(S) ≅ MF(S) and on its projectivization.

We consider the following equivalence relation on the space MF(S):

F ∼ tF, t > 0

Equivalently, we identify two quadratic differentials ϕ, ψ if ψ = t^{2}ϕ, t ∈ ℝ − {0}. The quotient

PMF(S) = (MF(S) − 0)/ ∼

is the space of projective classes of measured foliations. Similarly we define:

PML(S) = (ML(S) − 0)/ ∼

These spaces are homeomorphic to 𝕊^{−3χ(S)−n−1}, where n is the number of punctures. Here is a concrete way to embed PML(S) back to ML(S). Take a binding pair of simple closed geodesics α, β. We will consider α + β as a geodesic current using the counting transversal measures. Then by Lemma 12.20 the set

{λ ∈ ML(S) : p(λ) = i(λ, α + β) ≤ 1}

is compact. Thus the map

PML(S) ∋ [λ] ↦ λ/p(λ)

is a homeomorphic embedding. When convenient we will think of projective classes [λ] as elements of the “sphere”

{λ ∈ ML(S) : p(λ) = 1}

I recall now that for each pseudo-Anosov homeomorphism h : S → S we have a pair of projectively invariant measured foliations (F^{+}, F^{−}):

h(F^{+}) = kF^{+}, h(F^{−}) = k^{−1}F^{−}, where k > 1

Let λ^{+}, λ^{−} be the corresponding measured geodesic laminations . Thus, h has a pair of distinct fixed points on PML(S).

Theorem 12.37 (See [95].) For integers n > 0 the sequence of homeomorphisms

h^{n} : PML(S) → PML(S)

is convergent to the constant map [μ] ↦ [λ^{+}] uniformly on compacts in PML(S) − [λ^{−}].

Thus pseudo-Anosov homeomorphisms act on PML(S) the same way hyperbolic elements of Isom(ℍ^{n}) act on the ideal boundary ∂_{∞}ℍ^{n}. In particular this theorem included the statement that the foliations F^{±} are uniquely ergodic.

Exercise 12.38 Derive Theorem 12.31 of Marden and Strebel from Theorem 12.37.

We conclude with the following description of the laminations λ^{±} for a pseudo-Anosov homeomorphism h. Let γ be any simple closed geodesic on S, we regard γ as an element of ML(S). Consider the sequences

λ_{n}^{+} := h^{n}(γ), λ_{n}^{−} := h^{−n}(γ) ∈ ML(S)

The projective classes [λ_{n}^{+}], [λ_{n}^{−}] are convergent to [λ^{+}], [λ^{−}]. Fix any pair of binding simple closed geodesics α, β on S. The sequence of geodesic laminations λ_{n}^{±} is convergent to the geodesic lamination L^{±} underlying λ^{±} in the Chabauty topology; the number of points of intersection between L^{±} and at least one of the geodesics α, β (actually both) is infinite. Thus

lim_{n→∞} i(λ_{n}^{±}, α) + i(λ_{n}^{±}, β) = ∞

After normalization each sequence

λ_{n}^{±}/p(λ_{n}^{±})

is convergent to a measured lamination in the projective class [λ^{±}].

Proposition 12.39 The pair of measured laminations λ^{+}, λ^{−} is binding the surface S.

Proof: Suppose that this pair is not binding and μ is a measured lamination such that

i(μ, λ^{+}) + i(μ, λ^{−}) = 0

Recall that complementary regions C^{±} = S − λ^{±} are simply-connected polygons of finite area. Let ℓ be a leaf of μ: if it belongs to λ^{+}, then transversality of the foliations F^{+}, F^{−} on S implies that ℓ intersects λ^{−} transversally, hence i(μ, λ^{−}) > 0. Similarly we conclude that ℓ is not contained in λ^{−}. Thus it must be in one of the complementary regions. The leaf ℓ can not be closed, thus μ contains a sequence of leaves ℓ_{k} → ℓ which must eventually intersect both ∂C^{+}, ∂C^{−}. Contradiction. □

12.16 Thurston’s compactification of the Teichmüller space

Let S be a hyperbolic surface of finite area, S = ℍ^{2}/F, where F is a Fuchsian group. Let Ŝ denote the compactification of S where we add a circle to each cusp. We consider a real-analytic model

T_{R}(F) ⊂ Hom_{a}(F, P SL(2, ℝ))/P SL(2, ℝ)

of the Teichmüller space T (S) of the surface of S as in Section 5.4. Note that according to Theorems 9.4 and 9.32, the space T_{R}(F) is a component of the representation variety

Hom_{a}(F, P SL(2, ℝ))/P SL(2, ℝ)

which consists of discrete, faithful and admissible representations. The space of discrete and faithful representations of F to P SL(2, ℝ) is compactified by projective classes of small nontrivial minimal actions of F on trees. We know that the last are in 1-1 correspondence with projective classes of measured geodesic laminations in Ŝ. According to Theorem 12.28 this correspondence is a homeomorphism.

Thus we get a compactification of T (S) by points of PML(S). Let ∂_{∞}T (S) denote the set of points in PML(S) which are limits of sequences from T (S).

Projective classes of finite laminations are dense in PML(Ŝ). Suppose that γ is a simple closed geodesic on S, λ is the corresponding element of PML(S). Let δ denote another simple closed geodesic on S such that i(γ, δ) ≠ 0. One can construct a sequence of points in T (S) which is convergent to λ as follows. Let D_{γ} be the Dehn twist around γ, we will view it as an element of the mapping class group Mod_{S}. Then ψ_{n} = D_{γ}^{n}([S]) ∈ T (S) is a sequence of points, where [S] is the origin of T (S) represented by (S, id). The sequence ψ_{n} corresponds to a sequence of representations ρ_{n} : F → P SL(2, ℝ) given by the formula:

ρ_{n}(g) = D_{γ}^{n}(g)

Theorem 12.40 The sequence ψ_{n} converges to λ.

Proof: Splitting S along γ defines a decomposition A∗_{C}B of F = π_{1}(S) into an amalgamated free product (or HNN extension A∗_{C}) with amalgamation over C = ⟨γ⟩. Recall that the tree T dual to the lamination γ ⊂ S, is projectively equivalent to the simplicial tree T of the decomposition A ∗_{C} B (or A∗_{C}), the quotient of T by F is a segment or a circle with only one edge. For simplicity let me consider only the case of amalgamated free product decomposition. Note that the action of the elements D_{γ}^{n} on T (S) does not change marked hyperbolic structures on the components of S − γ. Let Q be the minimal F -tree which is the limit of the hyperbolic plane ℍ^{2} rescaled according to the sequence of representations ρ_{n}. Then the groups A and B have (distinct) global fixed points on Q and δ acts as a hyperbolic isometry (otherwise the tree Q would be trivial). Therefore (after some choice of metric on T) we get an isometric F -invariant embedding of the tree T into Q. Hence T = Q. □

This theorem implies that the points of accumulation of T (S) are dense in PML(S). We conclude that

∂_{∞}T (S) = PML(S)

This is called Thurston’s compactification of the Teichmüller space. It is known that T (S)∪ PML(S) is homeomorphic to the d-dimensional closed ball, where d is the (real) dimension of T (S).

Another approach to the description of T (S) ∪ PML(S) is due to Bonahon [25], we describe it below (for simplicity we restrict to the case of compact hyperbolic surfaces without boundary). Bonahon defines the Liouville measure L on M_{∞} as follows. Consider a product

[a, b] × [c, d] ⊂ 𝕊^{1} × 𝕊^{1} − Diag

of two circular arcs. Then the measure of its projection to M_{∞} is

L([a, b] × [c, d]/ ∼) =| log |\frac{(a − c)(b − d)}{(a − d)(b − c)}| |

This measure is invariant under the whole isometry group Isom(ℍ^{2}). Fix an origin S = ℍ^{2}/F in the Teichmüller space T (S) which we now consider as the space of marked hyperbolic structures on S. Then for each point q ∈ [X, f] ∈ T (S) we get an isomorphism ρ : F → F′ onto a Fuchsian subgroup of P SL(2, ℝ). This isomorphism is induced by a homeomorphism h : 𝕊^{1} → 𝕊^{1}. Let L_{q} be the pull-back of the measure L via h. Thus, we get an embedding

ι : T (S) ↪ GC(S), q ↦ L_{q}

Bonahon proves that for every L_{q} the self-intersection number i(L_{q}, L_{q}) = π^{2}|χ(S)|. Recall that ML(S) is the subset of GC(S) which is defined by

{α : i(α, α) = 0}

Each point q ∈ T (S) defines a length function

ℓ_{q} : C(S) → ℝ

as follows. If a representation ρ corresponds to q, then ℓ_{q}(γ) := ℓ(ρ(γ)) is the translational length.

The function ℓ_{q} extends continuously to ℓ_{q} : GC(S) → ℝ which coincides with i(L_{q}, •). In particular ℓ_{q}(γ) = 0 iff γ = ∅.

If α, β is a pair of binding simple closed geodesics on S then

lim_{j→∞} L_{q_{j}}/p(L_{q_{j}}) = λ ∈ ML(S)

if and only if the sequence of marked hyperbolic structures q_{j} is convergent to the projective class [λ] in T (S) ∪ PML(S). One way to see it is the following. Let L_{∞} denote the limit of L_{q_{j}}/p(L_{q_{j}}) (which exists up to a subsequence). Since i(L_{q_{j}}, L_{q_{j}}) = const and the intersection number is a continuous function, i(L_{∞}, L_{∞}) = 0 and hence L_{∞} ∈ ML(S). Each geodesic current σ is determined by the length function i(σ, •) : C(S) → ℝ. Then Theorem 11.13 implies that L_{∞} = λ.

13 Singular Hyperbolic Metrics and Pleated Surfaces

13.1 Singular hyperbolic metrics

Consider a smooth connected closed surface Ŝ and a finite collection of points

P := {p_{1}, ..., p_{m}}

on Ŝ, so that the punctured surface S := Ŝ − P has hyperbolic type. Let V ⊂ S be a finite collection of points. A “triangulation” T̂ of S based at V is a finite collection of smooth compact disjoint arcs in S with end-points in V so that:

1. No two arcs are isotopic in S (mod V).

2. No arc is isotopic to a point in S (mod V).

3. This system is a maximal system of arcs with the properties (1), (2).

Note that such “triangulation” does not have to be a triangulation in the usual sense. In our applications, V will consist of a single vertex. It follows from the property (3) that each puncture of S is contained in a region of Ŝ − T̂ which is either a monogon or bygon. Connect each puncture p_{j} with the vertices of this region by smooth arcs. The result is a completed triangulation T of S. Each complementary region of T is a triangle.

Now, fix a completed triangulation of S. Suppose that each triangle of this triangulation is given a complete hyperbolic metric of finite area and constant curvature −1. We assume that edges of each triangle are totally geodesic. Suppose that for every two triangles adjacent along an edge e, the length of e is the same in the both triangles. This collection of Riemannian metrics defines a path-metric on the whole surface S. We call such metric a singular hyperbolic metric if the following properties are satisfied:

• The total angle around each vertex V of the triangulation is at least 2π.

• The path-metric on S is complete.

Hence the universal cover X := S̃ of each singular hyperbolic surface is a CAT (−1)- space (see Section 4.2). There is a classification of isometries of X which is similar to the classification of isometries of ℍ^{n}: any nontrivial isometry γ of X:

• either has a fixed point in X (elliptic case);

• or has a single fixed point at infinity ξ = ξ_{γ} ∈ ∂_{∞}X and keeps its Busemann function β_{ξ} invariant (parabolic case);

• or it has a unique invariant geodesic A_{γ} ⊂ X (loxodromic case).

Suppose that γ ∈ Isom(X) is either loxodromic or parabolic. Consider the set

X_{ϵ} := {x ∈ X : ∃n ≠ 0|d(γ^{n}(x), x) ≤ ϵ}

Then convexity of the distance function in X implies that:

• If γ is loxodromic then X_{ϵ} is the union of the shortest geodesic segments between points of ∂X_{ϵ} and A_{γ};

• If γ is parabolic then X_{ϵ} is the union of geodesic rays emanating from points of ∂X_{ϵ} and asymptotic to ξ_{γ}.

In any case X_{ϵ} is contractible.

For each singular hyperbolic surface there is a multivalued developing map d : S − V → ℍ^{2} and a monodromy representation ρ : π_{1}(S − V) → Isom(ℍ^{2}). Note that completeness of S is equivalent to the fact that monodromy around each puncture is parabolic.

There is a combinatorial analogue of the Gauss-Bonnet formula for closed surfaces with singular hyperbolic metrics, which implies that area of any closed surface S with a singular hyperbolic metric is at most 2π|χ(S)|. Suppose that M is a complete hyperbolic 3-manifold and S is a surface with a singular hyperbolic structure. Then a continuous map f : S → M is called a singular pleated map, if:

• The restriction of f to any triangle in the completed triangulation of S is locally a totally-geodesic isometric embedding;

• f is proper on each triangle.

We will retain the name singular pleated map for the lift of f to the universal cover f̃: S̃ → ℍ^{3}.

We call the image of f a singular hyperbolic pleated surface. There is an analogue of the Kazhdan–Margulis’ lemma for fundamental groups of singular hyperbolic surfaces which have one singular vertex (in V) and a uniform upper bound on the total angle around this singular point. However we do not need it, since all singular hyperbolic surfaces that we shall consider admit pleated maps into complete hyperbolic 3-manifolds which induce monomorphisms of fundamental groups. Therefore, Kazhdan-Margulis’ lemma in ℍ^{3} implies the following property:

Property 13.1 If two loops on a singular hyperbolic surface S intersect and have length less than μ_{3}, then they generate a cyclic subgroup in π_{1}(S).

Similarly to the case of usual hyperbolic manifolds, it is true that for every μ_{3} > ϵ > 0 and singular hyperbolic surface S satisfying Property 13.1, the ϵ-thin part of S consists of the union of compact topological annuli and punctured topological disks. Let S_{ϵ}^{0} denote the complement to the union of punctured disks which are in the ϵ-thin part of S_{ϵ}.

Given a constant ϵ < μ_{3} and a singular hyperbolic surface S we collapse each component of S_{(0,ϵ]} to a point, the quotient ν_{ϵ}(S) has a natural path metric. Denote by χ(S) the Euler characteristic of S. By repeating arguments of the proof of Mumford compactness theorem (see for instance [1]) we prove:

Theorem 13.2 (Compare [24].) There is an universal function φ(x, y) such that: For any singular hyperbolic surface S (satisfying Property 13.1) and any two points a, b ∈ S_{[ϵ,∞)}, the distance d(ν_{ϵ}(a), ν_{ϵ}(b)) in ν_{ϵ}(S) is at most φ(χ(S), ϵ).

13.2 Existence theorem for singular pleated maps

Suppose that ρ : π_{1}(S) → Isom(ℍ^{3}) is a homomorphism and S is a singular hyperbolic surface. Recall this homomorphism is called parabolic if images of peripheral elements of π_{1}(S) are parabolic isometries. An element γ ∈ π_{1}(S) is called an accidental parabolic element for ρ if γ is not peripheral but ρ(γ) is parabolic.

Theorem 13.3 (Existence theorem for singular pleated maps.) Suppose that S = Ŝ − P is a surface of hyperbolic type (i.e. χ(S) < 0) and ρ : F := π_{1}(S) → Isom(ℍ^{3}) is a nonelementary parabolic representation. Then there exists a singular hyperbolic structure on S and ρ-equivariant singular pleated map f : S̃ → ℍ^{3}.

Proof: We shall prove this result under the following extra assumptions: ρ is discrete, faithful and has no accidental parabolic elements (i.e. the image of each nonperipheral loop is loxodromic). A general proof follows the same lines but is slightly more complicated (see [107], [72]).

Case 1. The surface S is compact.

There is a simple, homotopically nontrivial and nonperipheral loop γ on S.

Choose a point z ∈ γ. Then construct a triangulation of S which has a single vertex z ∈ V and γ is an edge of this triangulation. This triangulation lifts to a connected fundamental domain D for the action of F = π_{1}(S) in S̃. Denote vertices of D by z_{1}, ...z_{n}, they are preimages of z under the universal cover. Let e denote an edge of D emanating from z_{1} which covers the loop γ. Then an element g_{2} ∈ F corresponding to γ sends z_{1} to another vertex z_{2} of e. Let w_{1} = f(z_{1}) be any point on the axis of element ρ(g_{2}) ∈ P SL(2, ℂ), then we put f(z_{2}) = ρ(g_{2})(w_{1}) and extend the map f to the set of vertices of D in ρ–equivariant way. We assume that the image under f of any edge of the triangulation of D is a geodesic segment connecting corresponding points in ℍ^{3}. Extend f to each triangle of D by an affine map. Then we extend the map f in a ρ–equivariant way to the whole universal cover S̃. The pull-back of the hyperbolic metric from ℍ^{3} defines a singular metric on S̃ which is F –invariant. What is left to check is that this singular metric is hyperbolic. The image of the universal cover of the loop γ under f is a geodesic in ℍ^{3}. Therefore, the sum of angles of triangles adjacent to γ from the “left” is at least π, and the sum of angles of triangles adjacent to γ from the “right” is also at least π.

Case 2. Surface S is not compact.

Then we let V = ∅ and triangulate the surface S using the ideal vertices (from the set of punctures P) only. We let S = ℍ^{2}/F. The representation ρ is faithful and parabolic. Thus there is a natural ρ-equivariant bijection β = β_{ρ} between the set P̃_{F} of parabolic fixed points of the group F and the set P̃_{ρ(F)} of parabolic fixed points of ρ(F). The hyperbolic plane is tesselated by ideal triangles with vertices in P̃_{F} which are lifts of the triangles in our triangulation of S. For each 3 points a, b, c in P̃_{F} which span one of these triangles [abc] we take the ideal triangle [β(a)β(b)β(c)] ⊂ ℍ^{3}. We let f̃: [abc] → [β(a)β(b)β(c)] be a continuous map which is an isometry on the boundary geodesics. By adjusting boundary values we construct an equivariant pleated map f̃: ℍ^{2} → ℍ^{3}. □

13.3 Compactness theorem for pleated maps into hyperbolic manifolds

Fix a complete hyperbolic 3-manifold M, a compact B ⊂ M and a hyperbolic surface S of finite area. I recall that for a singular hyperbolic structure σ on S and μ ≤ μ_{3}, (S, σ)_{μ}^{0} denotes the complement to the union of punctured disk components of the μ-thin part of (S, σ).

Theorem 13.4 (W. Thurston) There exists a function θ(M, B, χ(S), μ) which has the following property. For any singular hyperbolic structure σ on S and any π_{1}-injective singular pleated map f : (S, σ) → M whose image intersects B and f_{∗} : π_{1}(S) → π_{1}(M) has no accidental parabolic elements, the diameter of (S, σ)_{μ}^{0} is at most θ(M, B, χ(S), μ).

Proof: Let μ = μ_{3}; M_{[μ/2,∞)}, M_{(0,μ/2]} be the μ/2–thick and μ/2–thin parts of M. We have the lower bound μ/2 on the minimal distance between different thin components of M. Denote by C the collection of cusps in M_{(0,μ/2]}. Collapse each thin component of M to a point, the quotient ν(M) has a natural induced path metric. Denote the projection ν : M → ν(M). Then for each r, the r-neighborhood of ν(B) can contain not more than j(r) elements of ν(M_{(0,μ/2]}), where j(r) = j_{M,B}(r) is a function which depends only on geometry of M and location of B. The singular pleated map f does not increase the distance, therefore f(S_{(0,μ/2]}) ⊂ M_{(0,μ/2]}. This means that f projects to a distance-nonincreasing map g : ν_{ϵ}(S) → ν(M) where ϵ = μ/2. According to Theorem 13.2, the diameter of ν_{ϵ}(S) is at most φ(χ(S), ϵ). Denote by {t_{1}, ..., t_{s}} the set of those points in ν(M_{(0,μ/2]} − C) which are within the distance at most φ(χ(S), ϵ) from ν(B). The number s is at most j(φ(g, ϵ)). These points correspond to solid tori in M_{(0,ϵ]}. Hence the injectivity radius over ν^{−1}({t_{1}, ..., t_{r}}) is at least η = η(M, B). Therefore the injectivity radius on S − f^{−1}(C) is at least η. On the other hand, all components of f^{−1}(C) ∩ (S, σ)_{μ}^{0} are homotopically trivial in S. This implies that the injectivity radius of (S, σ)_{μ}^{0} is least η. Thus, the diameter of (S, σ) is at most φ(χ(S), η) = θ(M, B, χ(S), μ). □

Corollary 13.5 (W. Thurston) Up to the precomposition with homeomorphisms of S there are only finitely many homotopy classes of maps from S into M which satisfy the hypothesis of Theorem 13.4.

Proof: Let me consider only the case of compact surfaces of genus g, the general case is similar. Fix a system of canonical generators

γ_{1}, ..., γ_{2g}

of F based at a point P ∈ S. Let f_{n} : (S, σ_{n}) → M be a sequence of maps as in Theorem 13.4. Since the diameter of (S, σ_{n}) is uniformly bounded, we may precompose f_{n} with a self–homeomorphisms of S to get generators g_{j} such that that the length of each based loop f_{n}(g_{j}) in M is at most 2θ(M, B, χ(S), μ). We may assume that f_{n}(P) ∈ B for each n. Therefore the corresponding sequence of representations ρ_{n} : F → G ⊂ P SL(2, ℂ) is relatively compact. Discreteness of G implies that this sequence contains only a finite number of different representations. □

Corollary 13.6 (W. Thurston) Let M be a closed hyperbolic 3-manifold, F ≅ π_{1}(S) is the fundamental group of a closed hyperbolic surface. Then there is only a finite number of copies of F embedded in Γ = π_{1}(M) up to conjugation in π_{1}(M) = G.

Proof: Take the compact B = M. Each embedding ρ_{n} : F → G is induced by a singular pleated map f_{n} : S → M whose image definitely intersects B. Then finiteness of the number conjugacy classes of images ρ_{n}(F) follows from Corollary 13.5. □

This corollary has admits some further generalizations: one can relax the assumption that F is the fundamental group of a closed hyperbolic surface and Γ is the fundamental group of a hyperbolic manifold, see [46], [182].

13.4 Geometrically tame ends of hyperbolic manifolds

In this section we shall study structure of ends of hyperbolic 3-manifolds with finitely generated fundamental groups, which are not geometrically finite. See Section 6.20 for definitions and notations related to ends of hyperbolic manifolds. Suppose that M is a complete hyperbolic 3-manifold with finitely-generated fundamental group, M^{0} := M_{ϵ}^{0} is the complement to the union of open Margulis cusps in M, M_{ϵ}^{c} is a compact core of M^{0}. We will consider only manifolds M which satisfy the following condition of Bonahon:

Condition B. The boundary ∂_{0}M_{ϵ}^{c} = ∂M_{ϵ}^{c} − ∂M_{ϵ}^{0} is incompressible. Equivalently, for every decomposition of G = π_{1}(M) into nontrivial free product A ∗ B there is a parabolic subgroup in G which is not conjugate into any of the free factors.

An end E of M_{ϵ}^{0} is geometrically finite iff E ∩C(Λ(G))/G is compact, where C(Λ(G))/G is the convex core of M. Another way to express it is to say that the subset E ⊂ M^{0} is precompact in the compactification M̄ = M ∪ Ω(G)/G. Note that geometric finiteness is clearly independent on the choice of compact core M_{ϵ}^{c}. In this section we will be interested in those ends of M which are not geometrically finite.

Definition 13.7 Suppose that E ⊂ M_{ϵ}^{0} − M_{ϵ}^{c} is an end of M, γ_{n} ⊂ E is a sequence of closed geodesics such that for each compact K ⊂ E only finite number of members of this sequence intersect K. Then the sequence {γ_{n}} is said to exit the end E. Similarly, a sequence of singular pleated maps f_{n} : S → P (E) exits E if f_{n}(S) ∩ E eventually leaves any compact subset of E. A geodesic ray ρ : [0, ∞) → E exits E if

lim_{t→∞} d(ρ(t), S) = ∞

Lemma 13.8 Assume that there is a sequence of closed geodesics γ_{n} ⊂ E exits the end E. Then the end E is not geometrically finite.

Proof: Suppose that E is geometrically finite. Then E ∩ C(Λ(G))/G is compact and for all large n ≫ 1 we have: γ_{n} ⊂ M − C(Λ(G))/G. However this means that the lifts γ̃_{n} ⊂ ℍ^{3} are not contained in the convex hull of the limit set. This contradicts to the property that the end–points of all geodesics γ̃_{n} are limit points of G. □

Theorem 13.9 (F. Bonahon [24]) Suppose that the end E is not geometrically finite. Then there exists a sequence of closed geodesics γ_{n} ⊂ E which exits the end E.

Proof: For simplicity I’ll restrict our discussion to the case when E = P (E), i.e. E is bounded by a closed surface S, see [24] for the general case. Let S_{n} ⊂ E be a surface homologous to S := ∂E so that d(S_{n}, S) ≥ n. Let E_{n} be the unbounded component of E −S_{n}. Consider the convex hull of S_{n} in E. Since E is not geometrically finite, the convex hull is unbounded. The convex hull of S_{n} is the union of convex hulls of finite subsets of S_{n}. However, for each finite subset X ⊂ S_{n} the convex hull C_{X} is within the distance at most 2 from the union of geodesics in C_{X} connecting points of X. Therefore, there is a sequence of geodesic segments g_{k} = [x_{k}, y_{k}] ⊂ E_{n} whose end–points belong to S_{n} so that length(g_{k}) → ∞ as k → ∞. We have:

d(y_{k}, x_{k}) ≤ C = diam(S_{n})

Denote by h_{k} the shortest geodesic segment connecting x_{k}, y_{k}, then length(h_{k}) ≤ C. Therefore for large k the sequence of broken geodesics g_{k} ∪ h_{k} determines a homotopically nontrivial loop q_{k}. For each n choose one of the loops q_{n} so that: length(q_{n}) → ∞ and q_{n} are homotopically nontrivial. Denote by g_{n} elements of G whose conjugacy classes represent q_{n}.

Up to a subsequence there are two possible cases:

(a) ℓ(g_{n}) ≥ ϵ > 0;

(b) ℓ(g_{n}) → 0 (recall that ℓ is the translational length of isometries of ℍ^{3}).

In the Case (a) we apply Lemma 4.9 and Lemma 4.10 to conclude that the closed geodesic γ_{n} ⊂ M freely homotopic to q_{n} is within (−4 log(ϵ/2) + 4)-neighborhood of q_{n}. Since d(q_{n}, S) → ∞ we conclude that the sequence of geodesics γ_{n} exits the end E.

Consider (b). Denote by δ the injectivity radius in the 1-neighborhood of the surface S. Then δ/2-Margulis tube (or cusp) U_{n} ⊂ M corresponding to the element g_{n} is disjoint from S. On the other hand, it contains a point within the distance ≤ −4 log(δ/2) from the loop q_{n} (Lemma 4.9). This means that all tubes U_{n} are contained in E. Since by our assumption the end E contains no cusps, we conclude that E contains a sequence of closed geodesics γ_{n} whose length tends to zero. Thus γ_{n} exits the end E. □

Definition 13.10 Suppose that E is an end of M and there is a sequence c_{n} of simple homotopically nontrivial loops in ∂_{0}E which corresponds to a sequence of closed geodesics γ_{n} in M, which exits the end E. Then the end E is called geometrically tame.

Let E be an end of M, S := ∂_{0}E; we will identify int(S) with a punctured surface.

Theorem 13.11 (F. Bonahon [24].) Suppose E is a geometrically tame end as above. Then there exists a sequence of singular pleated maps f_{n} : int(S) → P (E) which induce the same monomorphism of the fundamental groups as the embedding S ↪ E and such that {f_{n}(int(S))} exits E.

Proof: Using the same arguments as in Theorem 13.3 we construct a sequence of pleated maps f_{n} so that f_{n}(c_{n}) = γ_{n} (where c_{n} are simple closed loops on S). Suppose that there exists a compact K in M which intersects all members of a subsequence f_{n_{j}}(S). Then, according to the Compactness Theorem 13.4, the diameter of f_{n}(S)^{0} is bounded from above by some constant T independent on n. Therefore f_{n}(S)^{0} is contained in the T - neighborhood of K. However this contradicts the assumption that the sequence of geodesics γ_{n} on f_{n}(S) exits E. □

In these Notes we will rely heavily on the following

Theorem 13.12 (F. Bonahon [24]) All geometrically infinite ends of manifolds M satisfying the condition B, are geometrically tame.

Conditions of this theorem are satisfied in particular if M is homotopy–equivalent to a closed hyperbolic surface. End E of the manifold M^{0} is topologically tame if it admits compactification to a manifold with boundary.

Theorem 13.13 (F. Bonahon [24]) All geometrically tame ends of hyperbolic manifolds satisfying the condition B, are topologically tame. Thus, all such hyperbolic manifolds admit compactifications to manifolds with boundary.

Proposition 13.14 Let M be a complete hyperbolic manifold with finitely–generated fundamental group, E is an end of M. Then there exists a geodesic ray ρ ⊂ E which exits the end E.

Proof: Let x_{n} ∈ E be a sequence of points which exits the end E. Denote by [x_{n}, y_{n}] the shortest geodesic between x_{n} and the surface S ⊂ ∂_{0}M_{ϵ}^{c} which bounds the end E. Up to a subsequence, the sequence of points y_{n} is a convergent to a point y on the compact surface S and there is a limit of segments [x_{n}, y_{n}] in M which is an infinite ray ρ in M. We claim that this ray ρ exits the end E. Suppose that this ray intersects a compact ball B ⊂ M infinitely many times, i.e. there is a sequence of points z_{k} ∈ ρ ∩ B such that the distance between y and z_{k} along ρ tends to infinity. Then for each z_{k} we can find a point w_{k} ∈ [y_{n}, x_{n}] such that d(z_{k}, w_{n}) ≤ 1 and lim_{n→∞} length[w_{n}, y_{n}] = ∞. However d(w_{n}, y_{n}) ≤ dist_{H}(B, S)+1/2 is bounded from above. This contradicts the assumption that the segment [y_{n}, x_{n}] is the shortest between x_{n} and S_{n}. □

13.5 Ending laminations

Let G be a finitely generated Kleinian group satisfying Bonahon’s condition (B). Let Σ_{i} be the incompressible surfaces in M = M(G) corresponding to the geometrically infinite ends E_{i} of M. According to Bonahon’s theorem for each end E_{i} there is a sequence of simple closed loops γ_{n} ⊂ Σ_{i} which correspond to closed geodesics γ_{n}^{∗} ⊂ M so that ther sequence {γ_{n}^{∗}} “exits” the end E_{i}. Choose a complete hyperbolic metric of finite area on Σ_{i}. Consider the projection {[γ_{n}]} of the sequence of geodesic currents {γ_{n}} to P ML(Σ_{i}). Since P ML(Σ_{i}) the sequence {[γ_{n}]} has a convergent subsequence.

Theorem 13.15 Let [λ], [μ] be projective classes of measured laminations on Σ_{i} which are limits of subsequences in {[γ_{n}]}. Then

• λ, μ are maximal and connected.

• i(λ, μ) = 0.

Thus the support sets of λ, μ are equal.

Proof: See [24]. □

Definition 13.16 The geodesic lamination L equal to the support set of some (any) limit of a subsequence in {[γ_{n}]} is called the ending lamination of the end E_{i} and is denoted ϵ(E_{i}).

Remark 13.17 It seems to be unknown if the sequence {[γ_{n}]} has a unique limit in P ML(Σ_{i}) (it is possible that limit of a subsequence is not uniquely ergodic).

Conjecture 13.18 (Thurston’s ending lamination conjecture) The isometry class of the complete hyperbolic manifold M = M(G) is uniquely determined by the topology of M, the collection of ending laminations of the ends of M and by the conformal structure of Ω(G)/G. More precisely, suppose that G, G′ are discrete finitely generated subgroups of P SL(2, ℂ) satisfying Bonahon’s condition (B). Let ϕ : G → G′ be a type preserving isomorphism which is induced by a homeomorphism M(G) → M(G′) so that:

• For each component S_{i} ⊂ ∂M̄(G) the homeomorphism f : S_{i} → S′_{i} ⊂ M̄(G′) is induced by a conformal mapping.

• For each geometrically infinite end E_{j} of M̄(G) bounded by an incompressible subsurface Σ_{j} ⊂ M(G) the end E′_{j} of M(G′) corresponding to E_{j} under f is bounded by an incompressible subsurface Σ′_{j} = f(Σ_{j}) ⊂ M(G′) and f : Σ_{j} → Σ′_{j} carries the ending geodesic lamination ϵ(E_{j}) to ϵ(E′_{j}).

Then f : M(G) → M(G′) is homotopic to an isometry.

A partial progress towards this conjecture was achieved by Y. Minsky (see [146], [145]) who proved it in the case when the injectivity radius of M(G) is bounded from below and G is a surface group and in [147] where Minsky proves the ending lamination conjecture for the groups which are isomorphic to the fundamental group of a punctured torus via a type-preserving isomorphism. The results of [145] were generalized by M. Mitra [148] and E. Klarreich [117] who treated the case of freely indecomposable groups G so that M(G) has injectivity radius bounded from below.

13.6 Infinite coverings of hyperbolic manifolds

Theorem 13.19 (W. Thurston [192], R. Canary [35], K. Ohshika [161]). Let p : N → M be a covering between complete hyperbolic manifolds, where N has a finitely–generated fundamental group and satisfies Bonahon’s Condition (B). Suppose that E is a geometrically tame end of N. Then the following alternative holds:

(a) either M is finitely covered by a manifold M′ fibered over a circle and the group π_{1}(N) ∩ π_{1}(M′) is a surface group corresponding to the fundamental group of the fiber,

(b) or the restriction of p to E has finite multiplicity.

Remark 13.20 Dick Canary proved Theorem 13.19 under more general conditions instead of Bonahon’s condition (B).

Proof: Clearly it is enough to prove Theorem under the assumption that π_{1}(N) ≅ F is the fundamental group of an oriented complete hyperbolic surface S of finite area, and this isomorphism preserves the type of elements. (The surface S is naturally homeomorphic to a component of ∂_{0}N^{c} which bounds E.) We can also assume that S is distinct from the interior of a pair of pants. Suppose that f_{n} : (S, σ_{n}) → P (E) is a sequence of pleated maps which exits E, σ_{n} are singular hyperbolic structures on S. Let ρ : F → Γ = π_{1}(M) is the natural monomorphism.

According to the Compactness Theorem 13.4, up to a subsequence there are two possible cases:

Case 1. All images p(f_{n}(S)) intersect a compact K ⊂ M for all n.

Case 2. Singular pleated surfaces p(f_{n}(S)) eventually leave every compact subset of M. We start with the 1-st Case. Then, by Corollary 13.5, after taking a subsequence if

necessary we have a sequence of admissible automorphisms α_{n} : F → F and elements γ_{n} ∈ Γ so that the sequence of representations ρ_{n} := ad_{γ_{n}} ∘ ρ ∘ α_{n} : F ↪ Γ consists of the identically equal representations of F.

The sequence of outer automorphisms [α_{n}] ∈ Mod_{S} can not be finite since the pleated surfaces f_{n}(S) exit the end E and hence for each nonperipheral element γ ∈ F the length of the corresponding closed geodesic in (S, σ_{n}) tends to ∞ as n → ∞. We conclude that

γ_{n}F γ_{n}^{−1} = F

Thus γ_{n} belong to the normalizer N (F) of F in Γ and the projection of {γ_{n}} to N (F)/F is an infinite sequence, since [α_{n}] ∈ Mod_{S} is not subconvergent. Therefore it must be an element γ in the normalizer of F in Γ such that γ^{n} ∉ F for any n ∈ ℤ − {0} (see Corollary 5.11). Hence we get a subgroup Γ′ := F ⋊ ⟨γ⟩ ⊂ Γ which is isomorphic to the fundamental group of a mapping torus M_{α} of a homeomorphism of α : S → S. Note that the boundary of M_{α} has zero Euler characteristic, therefore Γ′ is a lattice in Isom(ℍ^{3}). This finishes the proof in the 1-st Case.

Case 2. Before giving a proof in this case I will describe an example which illustrates the situation. Suppose that M^{c} = M_{ϵ}^{c} is a Scott compact core of M_{ϵ}^{0}, p(S) ≅ S is a boundary component of ∂_{0}M^{c}, Γ = F ∗ ℤ, F ≅ π_{1}(S). Denote by E′ the end of M bounded by S. Then our conclusion would be that p(E) = E′ and the restriction of p to E is a 1-1. Assume that this conclusion is wrong. The manifold N has two ends E and D, so we assume that D is the end covering E′. Let g_{1} denote a generator of the factor ℤ. Denote g_{0} = 1, g_{1}^{2} = g_{2} ∈ Γ and F_{j} = g_{j}F g_{j}^{−1}, j = 0, 1, 2. So we can take a covering M′ over M whose fundamental group is F_{0} ∗ F_{1} ∗ F_{2} and M̃_{0} has at least 3 ends (covering E′) which correspond to the subgroups F_{j}. We denote these ends by E′, E_{1}, E_{2}. Retain notations M for this covering, Γ for its fundamental groups as well as for the covering p : N → M′ induced by the inclusion F_{0} → Γ. Let f_{n} : S → P (E) be a sequence of pleated surfaces which exits the end E. (See Figure 39.)

Figure 39: Pleated surface tries to exit two ends simultaneously.

We have two ends E_{1}, E_{2} of M which are bounded by surfaces S_{j} ⊂ M^{c} which are not homotopic to S. Then there exist complete geodesic rays ℓ_{j} ⊂ E_{j} which exit these ends. Lift these rays into rays R_{j} in N via the covering p. Then R_{j} must exit either the end E or the end D of N. However, they can not exit the end D since p(R_{j}) = ℓ_{j} does not intersect the surface S. So, each R_{j} exits the end E. This means that there exists an unbounded sequence of points x_{n} ∈ R_{1} ∩ f_{n}(S), y_{n} ∈ R_{2} ∩ f_{n}(S). Projections of these sequences to p(x_{n}), p(y_{n}) exit the ends E_{1}, E_{2} respectively. However the sequence of pleated surfaces pf_{n}(S) contains both p(x_{n}), p(y_{n}) , so it oscillates between two different ends E_{1}, E_{2}. This means that for each n the intersection p(f_{n}(S)) ∩ M^{c} is nonempty. Now Theorem 13.4 implies that p(f_{n}(S)) has uniformly bounded diameter and is contained in a compact subset of M. This contradicts the assumption that the sequences x_{n}, y_{n} on p(f_{n}(S)) exit the ends of M. This contradiction proves that p(E) = E′.

Now we consider the general situation. Our arguments follow [161]. Let p_{E} denote the restriction of the covering p to the end E of N^{0} bounded by the surface S^{0}.

Lemma 13.21 p_{E}^{−1}(p(S^{0})) ⊂ E is compact.

Proof: Suppose that the inverse image is not compact. Since p is a covering, p^{−1}p(S^{0}) intersects S^{0} by a finite collection of loops and arcs. Thus there are two possible cases. It may happen that p_{E}^{−1}(p(S^{0})) contains an unbounded connected component S_{1}^{0}, then S_{0}^{1} intersects all the singular pleated surfaces f_{n}(S) which exit the end E. Hence all singular pleated p(f_{n}(S)) intersect the compact set p(S^{0}), which is impossible. Thus we assume that p_{E}^{−1}(p(S^{0})) contains infinitely many compact components S_{n}^{0} disjoint from S^{0}. Fix an orientation on S^{0}. The oriented surfaces S_{n}^{0} determine relative homology classes [S_{n}^{0}] ∈ H_{2}(E, ∂_{P} E; ℤ). Note that these singular surfaces are π_{1}-injective, images of their fundamental groups have finite index in π_{1}(N). Hence all these cycles are nontrivial and are integral multiples of [S^{0}]. After taking a subsequence we can assume that [S_{n}^{0}] = k_{n}[S^{0}], k_{n} · k_{m} > 0; it’s enough to consider the case k_{n} > 0. Then there is a relative 3-chain σ_{1} ∈ C_{3}(E, ∂_{P} E; ℤ) such that ∂σ_{1} = k_{1}[S_{1}^{0}]−k_{2}[S_{2}^{0}]. Note that images of the classes p_{∗}[S_{i}^{0}] ∈ H_{2}(M, ∂_{P} M; ℤ) are equal to k_{i}[p(S^{0})], therefore ∂p_{#}(σ_{1}) = 0 ∈ C_{2}(M, ∂_{P} M; ℤ). Thus p_{#}(σ_{1}) is a 3-cycle which is nontrivial since p is a local diffeomorphism. This implies that M^{0} is compact, which again contradicts the assumption that we are in the Case 2. □

Clearly Lemma is valid for any choice of the surface S^{0} bounding the end E in S × ℝ. We conclude that the map p_{E} : E → M is proper. Let S̃ be one of the lifts of the surface S ⊂ N to M̃ = Ñ = ℍ^{3}; S̃ = ∂Ẽ, where Ẽ is a lift of E to ℍ^{3}. Suppose that the covering E → M does not have finite multiplicity, then there exists an infinite sequence γ_{n} ∈ π_{1}(M) such that:

• γ_{n}Ẽ ∩ Ẽ ≠ ∅;

• γ_{n}S̃ ∩ Ẽ = ∅;

• d(S̃, γ_{n}S̃) → ∞ as n → ∞.

Then S̃ ⊂ γ_{n}Ẽ for all n and hence γ_{n}^{−1}(S̃) ⊂ E. This implies that p_{E}^{−1}(p(S^{0})) is not compact. This contradiction concludes the proof of the Theorem. □

One of the standard applications of Theorem 13.19 is strong convergence of sequences of representations. Consider Theorem 9.39, Assertion (1) under the assumption that Λ(Γ) = 𝕊^{2}. Let G denote the geometric limit of the sequence ρ_{n}(F), lim ρ_{n} = ρ, ρ(F) = Γ. G is a discrete group which contains Γ. Thus we get a covering p : M(Γ) → M(G). Since Λ(Γ) = 𝕊^{2}, both ends of the manifold M(Γ) are geometrically infinite. Theorem 13.19 implies that either M(G) has finite volume or p is a finite covering. If p is finite then Theorem 9.12 implies that G = Γ and we are done. Otherwise vol(M(G)) < ∞ which contradicts the fact that infinite volume manifolds M(F_{n}) are convergent to M(G) in the quasi-isometric topology. □

14 Rips Theory

Consider a paired 3-manifold (M, P) with incompressible boundary ∂_{0}M; let W ⊂ M be the window of (M, P) and let G = π_{1}(M). We will assume that (M, P) satisfies the conclusion of Thurston’s geometrization theorem. We call an action G ↷ T on a metric tree elliptic (with respect to P) if the fundamental group of each component of the designated parabolic locus P has a global fixed point on T. The following theorem is crucial for the proof of the Hyperbolization Theorem as presented in our course:

Theorem 14.1 (J. Morgan and P. Shalen [152]). Suppose that T is a small minimal Gtree, so that the action G ↷ T is parabolic (with respect to P). Then the fundamental group of each component M_{j} of M − W has a global fixed point in T.

The original proof of this result is quite complicated and I must confess, I did not really try to understand it. In this Chapter I will give another proof of this result using the “Rips theory”. Our description of the “Rips theory” follows [21].

Roughly speaking the “Rips theory” is a device which for the given input:

⟨ a stable nontrivial action of a finitely-presentable group G on a metric tree T ⟩

produces the output:

⟨ a graph of groups decomposition of G ⟩

Definition of stable action is in the following section. Properties of the graph of groups decomposition of G are described in Section 14.13 (Theorem 14.73, Corollary 14.74). In Section 14.17 we derive Theorem 14.1 from Corollary 14.74.

14.1 Stable trees

Definition 14.2 (Stable subtrees) Consider a G-tree T and a nondegenerate subtree S ⊂ T. The subtree S is called stable if for each nondegenerate subtree S′ ⊂ S we have: F ix_{G}(S′) = F ix_{G}(S). Note that neither S nor S′ have to be G-invariant. If S is not stable then any subtree S′ above is called destabilizing subtree.

Example 14.3 If for every nondegenerate segment I ⊂ S we have: F ix_{G}(I) = {1} then S is stable.

Example 14.4 Suppose that S_{1}, S_{2} are stable subtrees in T and S_{1} ∩ S_{2} is nondegenerate. Then S := S_{1} ∪S_{2} is also a stable subtree in T. Indeed, let S′ ⊂ S be a destabilizing subtree, g ∈ G is an element in F ix_{G}(S′) − F ix_{G}(S). Without loss of generality we can assume g ∉ F ix_{G}(S_{1}). Since S_{1} is stable we conclude that S′_{1} := S_{1} ∩ S′ is a single point. Hence S′ ⊂ S_{2} and stability of S_{2} implies that g|S_{2} = id. Therefore g ∈ F ix(S_{1} ∩ S_{2}) − F ix(S_{1}) and by stability of S_{1} we get: S_{1} ∩ S_{2} is either empty or a single point which contradicts our assumptions.

Clearly if S_{1} ⊂ S_{2} are nondegenerate trees and S_{2} is stable then S_{1} is also stable. Hence each stable subtree S ⊂ T is contained in a unique maximal stable subtree M. In particular, if two distinct maximal stable subtrees intersect then the intersection is degenerate.

Exercise 14.5 Construct example of a G-tree T with two distinct maximal stable subtrees M_{1}, M_{2} ⊂ T.

Definition 14.6 A nontrivial action of a group G on a tree T is called stable if every nondegenerate subtree S ⊂ T contains a stable subtree.

Example 14.7 Consider a simplicial tree T (where each edge has unit length) and let G be a subgroup in the group of automorphisms of T. Then the action of G on T is stable. Indeed, if S is any nondegenerate subtree of T then S contains a proper nondegenerate subsegment I of an edge of T. This segment I is a stable subtree: if g ∈ G fixes a nondegenerate subsegment I′ ⊂ I then g fixes I as well.

To get an example of unstable action consider T := Cone(ℍ^{2}), G := Isom(T). Then for each nondegenerate arc I ⊂ T there is an element g ∈ G and a subarc I′ ⊂ I such that g|I ≠ id, g|I′ = id.

Exercise 14.8 Construct an example of a finitely-generated group G and an unstable action of G on a tree.

Theorem 14.9 Suppose that G is a nilpotent subgroup of π_{1}(M), where M is a Haken 3-manifold (satisfying the conclusion of Thurston’s geometrization theorem). Then G is finitely generated.

Proof: We first note that if Γ is a torsion-free nonebalian finitely generated nilpotent group then cd(Γ) ≥ 3. Thus either M is a Nil-manifold and G is a finite index subgroup iin π_{1}(M) or G is abelian. From now on we shall assume that G is not finitely generated abelian group. It was proven in [57] that if G is abelian and not finitely generated then G is isomorphic to the additive group ℚ of rational numbers.

Consider the Jaco-Shalen-Johannson decomposition of M into the components M_{j}. This decomposition gives rise to an action of π_{1}(M) on a simplicial tree T (this is an action without inversions). We give T metric structure by assuming that each edge has unit length. The decomposition of M has the peroperty: if γ ∈ π_{1}(M_{j}) − {1}, then any element of the centralizer of γ in π_{1}(M) belongs to π_{1}(M_{j}). The group G acts on the tree T, thus G consists of elliptic automorphisms of T iff a single nontrivial element γ ∈ G is elliptic. If such element exists, then it fixes a vertex of T and hence (up to conjugation) we can assume that γ ∈ π_{1}(M_{j}) for some component of the decomposition of M. Hence G ⊂ π_{1}(M_{j}). Either the manifold M_{j} is a Seifert manifold, or it is an interval bundle over a surface or π_{1}(M_{j}) is a discrete subgroup of SO(3, 1), in both cases we immediately get a contradiction. Thus we assume that each nontrivial element γ ∈ G acts on the simplicial tree T as a hyperbolic automorphism, however this implies that ℓ_{T} (γ) ≥ 1. This means that one can extract only finitely many roots from γ and hence G is finitely generated. □

Proposition 14.10 Let T be a nontrivial G-tree. Then each of the following conditions implies stability of the action of G on T:

1. The set of arc stabilizers satisfies the ascending chain condition: if G_{1} ⊂ G_{2} ⊂ G_{3} ⊂ ... is a chain of arc stabilizers then G_{i} = G_{i+1} for all but finitely many i.

2. For each arc I ⊂ T we have: F ix_{G}(I) is trivial.

3. The action of G is small and G is the fundamental group of a compact Haken manifold or G is a discrete group of isometries of a complete simply-connected manifold X whose sectional curvature is pinched between two negative constants (for instance X = ℍ^{n}).

4. If s_{1} ⊃ s_{2} ⊃ ... is a sequence of arcs in T converging to a point then F ix(s_{i}) = F ix(s_{i+1}) for all but finitely many i.

Proof: (1) Let S ⊂ T be a subtree, then (by (1)) S contains a subarc J such that F ix_{G}(J) = F ix_{G}(J′) for any subarc J′ ⊂ J. Hence J is a stable subtree in S.

(2) Obvious.

(3) First we note that every finitely generated virtually nilpotent group satisfies the ascending chain condition. Since the action of G is small, arc stabilizers are virtually nilpotent subgroups in G. If G is the fundamental group of a Haken manifold then each virtually nilpotent subgroup of G is finitely generated (see Theorem 14.9). In the second case we apply [27] to prove that each virtually nilpotent subgroup N in G is finitely-generated. In both cases, if N_{1} ⊂ N_{2} ⊂ ... is a chain of virtually nilpotent subgroups in G, then N_{i} = N_{i+1} for all sufficiently large i and we can apply (1).

(4) This assertion is analogous to (1) and we leave it to the reader. □

The above proposition explains the name stability: sequences of stabilizers are eventually stabilizing. Note however that smallness of the action alone does not imply that the action is stable: Dunwoody [50] constructed example of a finitely generated group G and a small but unstable action of G on a tree T.

14.2 Unions of bands and band complexes

A general G-tree T is a complicated dynamical system, the goal of this section is to describe corresponding finite combinatorial object which encodes the information about the action of G on T. Roughly speaking a union of bands is a singular one-dimensional foliation with transversal-invariant measure. Generic fibers of this foliation do not have to be 1-manifolds, they are 1-dimensional graphs.

We start with a real graph, i.e. finite simplicial^{4} 1-complex Γ where each edge is identified with an interval in ℝ. We orient each edge of Γ.

Remark 14.11 This definition does not require existence of a global metric on Γ whose restriction on edges e of Γ is consistent with embeddings e ↪ ℝ. For instance the triangle with the sides of the length 1, 1, 3 is an example of a real graph. However we do have a metric on each edge of Γ. Given these metrics we construct the induced path-metric on Γ.

Let I := [0, 1]. A band B is the product b × I where b is a compact nondegenerate segment of ℝ. The segment b will be identified with b × {0}, which is called the bottom of B. The product b × {1} will be called the top of B. We will use the name bases for the top and bottom segments of B. We have the reflection

δ_{B} : B → B, δ_{B}(s, t) = (s, 1 − t)

If S is a subset of one of the bases of B then δ_{B}(S) is called the dual of S.

Subsets of b × {t} (for t ∈ I) and {s} × I (where s ∈ b) will be called vertical and horizontal subsets of B respectively. We will think of B as a fiber bundle over b with the vertical fibers {s}×I. Take a finite collection B_{i} := b_{i} ×I (i = 1, ...n) of bands and isometric embeddings

f_{i}, f′_{i} : b_{i} × {0}, b_{i} × {1} → Γ, i = 1, ...n

whose images are contained in closed edges of Γ. However we do not require images of these maps to be equal to edges of Γ.

Definition 14.12 A union of bands is the quotient space Y of the disjoint union

Γ ⊔ B_{1} ⊔ ... ⊔ B_{n} = γ ⊔ B

where we identify points of bases with their images in Γ. We will identify each band with its image in Y. We require each point of Γ to belong to the image of at least one base.

Define bases in Y as images of bases of B_{j} to Y. We retain the notation b for bases in Y. We define the boundary ∂Y as the union of images of the vertical boundary segments of the bands B_{i} in Y.

Remark 14.13 The union of bands Y is not just a topological space but we retain the information about the geometry of Γ, bands, attaching maps and product decomposition of the bands.

Each union of bands Y is foliated, the leaves of Y are are equivalence classes generated by the equivalence relation:

x ∼ y if x, y belong to a vertical fiber of a band

Thus each leaf is a union of vertical fibers in bands. Each leaf has a natural topology: a subset is open iff its intersection with any vertical fiber of any band is open. Thus every leaf is is homeomorphic to a locally finite 1-dimensional CW-complex.

The reader can think of Y as a foliated space with a flat connection (given by horizontal segments) and a metric structure transversal to the foliation.

Definition 14.14 A subset of Y is called horizontal if it is contained in Γ or is contained in a single band and is horizontal there.

4It will become clear in the Definition 14.16 that we can not work with general finite graphs.

Figure 40 gives an example of a union of bands. In this case the union of bands is a surface with boundary and the foliation is a foliation by trajectories of a holomorphic quadratic differential on this surface. Generic leaf of this foliation is a 1-manifold, however there are several trajectories which branch, classically they are called singular (or critical) trajectories; there are three singular points p, q, r, each singular point is located on the boundary of the surface.

Figure 40: Example of a union of bands. Identify the curves with the labels x_{i}y_{i} to get the band B_{i}, i = 1, 2, 3.

Since for general unions of bands we can not expect generic leaves to the 1-manifolds, the definition of singular leaf has to be modified. One way to do it is to declare a leaf to be nonsingular iff it is disjoint from vertical boundary fibers of any band, this is the approach of [71]. It is clear however that this definition is not equivalent to the classical one in the case of product foliation of a compact annulus by concentric circles: the boundary leaves are obviously noncritical. We will give another definition:

Definition 14.15 A leaf L is called nonsingular if it can be exhausted by compact subgraphs C_{j} so that the following is satisfied:

Let C_{j}^{o} be the interior of C_{j} in L. Then there is an open neighborhood U ⊂ Y of C_{j}^{o} such that U is naturally homeomorphic to an interval bundle over C_{j}^{0} where the interval is either open or half-open. This interval bundle has a flat connection whose leaves are contained in in leaves of Y and fibers of the interval bundle are horizontal in Y.

Otherwise L is called singular.

In the next section we will give an alternative definition in terms of pushing in Y.

Definition 14.16 (Measures of complexity of a union of bands.) Define the height η(Y) of a union of bands Y to be the total number of bands in Y.

For each base b of a band B in Y define the weight w(b) as follows:

• If B is an annulus (i.e. f_{dual(b)} = f_{b} ∘ δ_{b}) we let w(b) = 0.

• If B is a Moebius band we let w(b) = 1/2.

• Otherwise w(b) = 1.

Define weight w(z) of a point z ∈ Γ to be the sum of weights of bases containing z. A block γ in Y is the closure of a component of the union of interiors of bases of Y.^{5} The complexity σ(γ) of a block γ equals

Complexity(γ) = σ(γ) = max{0, −2 + ∑_{b is a base in γ} w(b)}

The complexity σ(Y) is the sum of complexities of blocks in Y.

−2 in the definition of complexity does not look natural at the first glance, however it is crucial for proving that complexity does not increase under the Processes I and II of the Rips machine.

Exercise 14.17 Give a complete list of all blocks of zero complexity which do not contain bases of zero weight.

14.3 Pushing

Let S, S′ ⊂ Y be horizontal subsets and p be a path in a leaf of Y.

Definition 14.18 The subset S pushes into S′ along the path p if there is a homotopy H : S × I → Y so that:

• H(x, 0) = x for each x ∈ S, H(S, 1) ⊂ S′.

• H(S, t) is horizontal for each t ∈ I.

• H(z × I) is contained in a leaf of Y for each z ∈ S.

• There is a point z_{0} ∈ S such that H(z_{0}, t) = p(t), t ∈ I.

We say that S pushes onto S′ along the path p if it pushes into S′ along the path p and H(S, 1) = S′.

Given a point z ∈ Y consider the maximal connected horizontal subset h(z) ⊂ Y containing z. Clearly h(z) is a finite graph which is a horizontal segment of a band if z ∉ Γ. The finite graph h(z) has a natural path-metric induces from the metric on edges of Γ and the metric on horizontal slices of bands. Given ϵ > 0 we define horizontal neighborhoods N(z, ϵ) of points z ∈ Y as closed ϵ-neighborhoods of z in h(z).

Given z ∈ Y define P (z) as the collection of points y ∈ Y so that a horizontal neighborhood of z pushes onto a horizontal neighborhood of y along a path

p : I → a leaf of Y which connects z to y

Note that y ∈ P (z) iff z ∈ P (y).

Remark 14.19 Note that our definition of P (z) is different from [21].

Definition 14.20 A subset S of Y is called pushing-saturated if given any leaf L of Y, a path p : I → L and ϵ > 0 such that N(z, ϵ) pushes along p, it follows that p(1) ∈ S.

Note that P (z) is the minimal pushing-saturated subset of Y containing z. In other words, S ⊂ Y is pushing-saturated iff S ⊃ P (z) for each z ∈ S.

5Since Γ is a simplicial complex it follows that each block is an interval.

Exercise 14.21 Show that the closure and accumulation set of any pushing-saturated set are again pushing-saturated. Unions and intersections of pushing-saturated sets are again pushing-saturated.

Exercise 14.22 A leaf is singular iff it contains a proper pushing-saturated subset. Equivalently, a leaf L is non-singular iff L = P (z) for each z ∈ L.

Definition 14.23 A union of bands is minimal if every pushing-saturated subset of Y is dense in Y and simplicial if every leaf of Y is compact.

At the first glance this definition does not look very natural: why do we define minimality using pushing-saturated sets instead of simply saying that every leaf is dense? The main reason for that is the way the Rips Machine works. The Machine operates not with the leaves L of Y but rather with unions of neighborhoods U_{j} = U(C_{j}^{0}) of finite subgraphs C_{j}^{0} which exhaust L as in the Definition 14.15. One of the features of the Machine is step-by-step elimination of certain neighborhoods U_{j} from the union of bands. Such neighborhoods U_{j} exist only for pushing-saturated subsets and not for arbitrary leaves.

Definition 14.24 Suppose that W is a compact proper and pushing-saturated subset of a leaf L of Y. Then W is called a fault.

It is clear that each leaf of Y is pushing-separated. Below we give a necessary condition for a leaf L of Y to contain a proper pushing-saturated subset (see Figure 41):

(a) Either the leaf L contains a local cut point, i.e. there is a point z ∈ L ∩ Γ which locally separates the union of Γ and those bands that contain z.

(b) Or the leaf L contains a point z ∈ L ∩ Γ such that: z is an end-point of a base b and an interior point of a base b′ (such point z will be called a dead end).

Figure 41: Critical points of leaves.

Point z as in (a) or (b) is called a critical point of L. Note however that existence of a critical point is only necessary but not sufficient condition for L to be singular.

Exercise 14.25 Prove that:

• Each singular leaf contains only finitely many critical points;

• If L′ ⊂ L is a proper pushing-saturated subset of L then the frontier of L′ in L consists of critical points. In particular, L contains only finitely many pushing-saturated subsets and Y contains only finitely many singular leaves.

14.4 Transversal measure on union of bands

Let Y be a union of bands, the goal of this section is to define a transverse measure on Y (with respect to the foliation of Y into vertical leaves). Let g : I → B be a continuous map of the unit interval into a band B = b × I. Then we say that g is transverse to the vertical foliation of B if the composition π ∘ g is injective where π : B → b is the projection to the first factor. If g is transverse then we define the transverse measure μ(g) as the length of the subinterval π(g(I)) ⊂ b. Suppose now that g : I → Y is a cellular map and we have a finite subdivision of I into subintervals I_{j} so that for each j the image g(I_{j}) is contained in a single band B_{j} ⊂ Y. Then we say that g is transverse to the vertical foliation of Y if each restriction g|I_{j} is transverse to the vertical foliation of B_{j}. For transversal paths g we define the transverse measure μ(g) of g as the sum

∑_{j} μ(g|I_{j})

This definition extends to cellular paths g such that I has a finite subdivision into subintervals I_{j} so that for each j either g|I_{j} is transverse to the vertical foliation of B_{j} or is contained in a leaf of B_{j}, in the latter case we let μ(g|I_{j}) := 0.

14.5 Imanishi’s Theorem

Theorem 14.26 (Imanishi [94], “Dynamical decomposition of unions of bands”.) Each union of bands Y contains only finitely many faults. Suppose that Y has no faults at all. Then Y splits into disjoint union of components each of which is either minimal or simplicial. Each simplicial component is an interval bundle over a leaf in that component.

Proof: The first assertion follows from the Exercise 14.25. Our proof of the second assertion follows [71]. Consider a finite regular leaf L. Thus (by definition of regularity) all leaves nearby are also finite and regular, let U(L) be the maximal connected horizontal neighborhood of L which consists of finite regular leaves. Clearly U(L) is an interval bundle over L. The frontier of U(L) in Y consists of finite singular leaves. Clearly each finite singular leaf contains a fault, hence U(L) is a component of Y, by compactness there are only finitely many such components, let U denote their union.

Pick a component M of Y − U. Our goal is to show that M is minimal, i.e. each pushing-saturated subset L of M is dense in M. Without loss of generality we may assume that L is contained in a single infinite leaf.

Step 1:

Proposition 14.27 There is a number δ > 0 (depending on L) such that for every base b, every component in b − cl(L) has length at least δ. In particular no infinite leaf closure in M intersects a base in a Cantor set.

Proof: Without loss of generality we can assume that L ∩ b is an infinite set. Choose δ as follows: if x is an end-point of a base c then the distance between x and int(c) ∩ L is either zero or at least δ.

Suppose that J = [x, u] is a component of cl(L) ∩ b which has length λ < δ. Since the accumulation set of a pushing-saturated set is also pushing-saturated we conclude that for each end-point x of J the set P (x) has infinite length and intersects b infinitely many times. Thus we can find two distinct points y, z ∈ P (x) ∩ b which satisfy:

• |y − z| < λ; • a nondegenerate subinterval [x, w] of J pushes^{6} into the interval [y, z] along a vertical

path p connecting x to y.

Define x′ ∈ [x, u] by |x − x′| = |y − z| < λ. Note that x′ ∉ cl(L), hence the interval [x, x′] does not push into [y, z] along p. The path p is a finite composition of vertical paths:

p_{1} ∗ .... ∗ p_{k}

each contained in a single band. Hence we can find j ≥ 1 so that [x, x′] pushes along p′′ := p_{1} ∗ ... ∗ p_{j−1} and does not push along p′ := p_{1} ∗ ... ∗ p_{j}. (If j = 1 then we set p′′ be the constant path.) Let x′′ ∈ (x, x′] be the point closest to x such that [x, x′′] does not push along p′. Then:

• the point p′′(x′, 1) ∈ Γ is not in cl(L);

• p′′(x′, 1) is an end-point of one of the bases c (otherwise we can continue pushing along p′);

• 0 < d(c, L) = λ < δ (since the whole open interval (p′′(x, 1), p′′(x′, 1)) is the image of (x, x′) under pushing along p′ and thus is disjoint from cl(L)).

This contradicts our choice of δ and hence proves Proposition 14.27. □

Step 2: Consider the intersection b ∩ cl(L) of a base b ⊂ M ∩ Γ. Without loss of generality we may again assume that this intersection is infinite. Pick a pushing-saturated subset K = P (x) for a point x of the frontier of b ∩ cl(L) in b; we may choose x an interior point of b, let J ⊂ b denote an interval with end-point x such that int(J) ∩ cl(L) = ∅. Then K has infinite length, let δ = δ(K) be given by the Proposition 14.27. Choose y, z ∈ K such that:

• 0 < |y − z| < δ;

• If p is a path in K connecting x to y then p pushes a nondegenerate subinterval [x, x′] ⊂ J into the interval [y, z].

Then [y, z] is contained in cl(K) ⊂ cl(L) (by Proposition 14.27). On the other hand, an interval [x, x′] pushes into [y, z] and (x, x′] ∩ cl(L) = ∅. This is a contradiction. □

Corollary 14.28 Y is minimal iff each leaf of Y is dense and Y contains no compact proper and pushing saturated subsets.

14.6 Band complexes

Note that each union of bands is homotopy equivalent to a finite graph. Thus unions of bands clearly do not suffice to describe actions of groups on trees (unless these groups are free groups). So to get more complicated finitely presented groups we have to add 2-cells to unions of bands.

Definition 14.29 A band complex X based on a union of bands Y is a finite 2dimensional CW-complex obtained by taking Y and attaching 0, 1 and 2-cells as follows:

• The 1-cells of X intersect Y in a subset of Γ.

• Each component of the intersection of Γ and a 2-cell of X is a point.

Figure 42: A band complex.

• Each component of the intersection of a band of Y and a 2-cell of X is vertical.

We define leaves in a band complex X (based on a union of bands Y) as leaves in Y. Similarly, X is called minimal (resp. simplicial) if Y is minimal (res. simplicial). Complexity σ(X) and height η(X) are defined as σ(Y) and η(Y) respectively. Define the transversal measure on X as follows.

A cellular path g : I → X is said to be transverse to the vertical foliation of X if I can be subdivided into finite number of subintervals I_{j} so that for each j either:

(a) g|I_{j} : I_{j} → Y is transverse to the vertical foliation of Y (see Section 14.4) in which case we let μ(g|I_{j}) be the transverse measure of g|I_{j} : I_{j} → Y, as defined in Section 14.4;

(b) or g_{j}(I_{j}) is contained in the closure of X − Y, in which case μ(g|I_{j}) := 0. For transverse paths g we define the transverse measure μ(g) as the sum

∑_{j} μ(g|I_{j})

It is clear that if g : I → X is a path such that g(0) = x_{0}, g(1) = x_{1} then we can homotope g (rel. {0, 1}) to a a path transverse to the vertical foliation of X.

Definition 14.30 Let X̃ → X be a cover of X (usually but not always it will be the universal cover). The transverse measure lifts from X to X̃. A generalized leaf in X̃ is an equivalence class for the equivalence relation x ∼ x′ iff there is a path in X̃ connecting x to x′ whose transversal measure equals zero. Then the leaf space X̃/∼ of X̃ is the quotient of X̃ by the above equivalence relation. The leaf space is equipped with a pseudometric

d([x], [x′]) := inf{μ(g) : g is a path in X connecting x to x′}

We will say that the pair (X, X̃) is regular if the leaf space (T_{X}, d) is a metric tree.

Example 14.31 Band complexes corresponding to interval exchange transformations. Let E_{r,σ,χ} be a generalized interval exchange transformation (defined in Section 12.12).

We let Γ be the real graph [0, ρ]. Let B_{i} := [x_{i−1}, x_{i}] × I be the bands, i = 1, ..., n, define the attaching maps for these bands as id = f_{i} : b_{i} = [x_{i−1}, x_{i}] → [x_{i−1}, x_{i}] ⊂ Γ and f′_{i} = λ_{i} : b_{i} × {1} → Γ. As the result we get a union of bands Y associated with the interval exchange transformation E_{r,σ,χ}. It is clear that the total space of Y is homeomorphic to a surface with boundary S. We let X be the band complex obtained by attaching a 2-cell to each boundary component of S so that each attaching map is a homeomorphism. (More generally, one can attach cells via degree maps of degrees ≥ 1, in which case the fundamental group of the resulting band complex is the fundamental group of a 2-dimensional orbifold.)

6Up to this point all our arguments were valid for leaves (instead of pushing-saturated sets). However to get the interval [x, w] we have to use pushing.

Below we use interval exchange transformations to give two examples of band complexes: one when d is a metric and the second when d is identically zero, but X̃/ ∼ is not a single point.

Example 14.32 A band complex X so that the pseudometric d associated to the universal cover of X is a metric and (X̃/∼, d) is a metric tree.

Start with a closed hyperbolic surface S and a singular measured foliation F on S associated with a holomorphic quadratic differential on S. Then using the construction of interval exchange transformation in Section 12.12, we get a union of bands Y obtained by splitting S along several arcs contained in the leaves of F. We recover S from Y by attaching 2-cells along loops in Y corresponding to these arcs. We get a band complex X. Let X̃ be the universal cover of X. Then (X̃/∼, d) is isometric to the tree T_{F} dual to the lift of F to X̃.

□

Example 14.33 A band complex X so that the pseudometric d associated to the universal cover of X is identically zero.

Let T^{2} be a flat 2-torus, F a transversally measured geodesic foliation on T^{2} which is the stable foliation of f. Let τ : T^{2} → T^{2} be the Cartan involution which fixed a point O ∈ T^{2}. Clearly F is invariant under τ. The quotient T^{2}/⟨τ⟩ is the 2-sphere S^{2} and the transversally measured foliation F projects to a measured foliation F̄ on S^{2} with four points removed (these are the projections of the points fixed by τ). We next associate an interval exchange transformation to the foliation F̄.

We describe the quotient foliation F̄ in Figure 43: the torus T^{2} is obtained by identifying sides of the fundamental rectangle R, the foliation F corresponds to the vertical geodesic foliation on R; the orbifold T^{2}/⟨τ⟩ is obtained by making identifications on sides of the triangle Δ (see Figure 43). In Figure 44 we describe the band complex X corresponding to the foliation F̄ on S^{2}.

To obtain X we split S^{2} open along seven disjoint segments each of which is contained in a leaf of F̄ and attach 2-cells to each boundary circle of the resulting surface Σ by homeomorphic maps of the circles.

The surface Y = Σ is described in the Figure 44: we identify segments with the labels [px], [pq], [qm], [my] in the union of bands, the foliation F̄ on Y is the “vertical” foliation of each rectangle B_{j}, j = 1, ..., 6. To get X from Y attach 2-cells indicated by shaded regions in the Figure 44. The dotted arcs connecting the points p, x, y, m to themselves are to be contracted to points.

Since each leaf of F is dense in T^{2} the same is true for F̄. The sphere X is simply connected, hence the universal cover X̃ is the same as X itself. We conclude that the leaf space X̃/ ∼ is not Hausdorff (each point is dense) and the pseudometric d on X̃/ ∼ is identically zero. On the other hand, X̃/∼ consists of continuum of points. □

Let X be a band complex, X̃ is a cover of X, then we shall use the notation T_{X} for the quotient X̃/∼ (we will supress the choice of X̃. Let q : X̃ → T_{X} denote the projection.

Definition 14.34 Let G be a finitely presented group acting on a tree T. A resolution for this action is a G-equivariant map

ρ : X̃ → T

where X̃ is cover of a band complex X such that π_{1}(X)/π_{1}(X̃) = G and:

• For each generalized leaf L ⊂ X and a component L̃_{0} of its lift to X̃ the image ρ(L̃_{0}) is a point.

• For each base b ⊂ Y ⊂ X and a component b̃_{0} of its lift to X̃ one can subdivide b̃_{0} into finitely many subintervals I_{j} so that ρ|I_{j} is an isometric embedding for each j.

The band complex X is said to be resolving the tree T. Resolution ρ is called exact if for each point t ∈ T the preimage ρ^{−1}(t) is path connected.

Clearly, if (X, X̃) gives an exact resolution of T then (X, X̃) is necessarily regular and the generalized leaves of X are path connected. Below construct resolutions which are adapted to a collection of finitely generated subgroups F_{i} of G:

Theorem 14.35 Suppose that G is a finitely presentable group, T is a G-tree and F_{1}, ..., F_{q} are finitely generated subgroups of G each of which fixes a point on T (q might be equal). Then T has a resolving regular band complex X such that X̃ is the universal cover of X and for each F_{j} there exists a connected subgraph D_{j} ⊂ ∂Y such that F_{j} is contained in the image in G of π_{1}(D_{j}).

Proof: Let ⟨g_{1}, ..., g_{n}|R_{1}, ..., R_{k}⟩ be a finite presentation for the group G where the list of generators includes all generators of the groups F_{1}, ..., F_{q}. Let Z = {z_{1}, ..., z_{q}} ⊂ T be the finite collection of fixed points of the groups F_{1}, .., F_{q}. Pick a point x ∈ T and consider a finite subset S ⊂ T which consists of:

• {x} ∪ Z and images of {x} ∪ Z under g_{j}^{±1}, j = 1, ..., n;

• images of x under all subwords in the relations R_{i}, i = 1, ..., k.

Let K be any finite compact subtree in T which contains the set S, for instance we can take the convex hull of S as the subtree K. Consider the collection

ϕ_{j} : T_{j} := K ∩ g_{j}^{−1}K → T′_{j} := g_{j}K ∩ K

of the actions of g_{j}, j = 1, ..., n. The subtrees T_{j} cover K (because of the first condition on S). We introduce structure of a simplicial complex on K as follows: the set of vertices consists of {x} ∪ Z and vertices of K. Each edge of the resulting simplicial complex has length induced from K. Finally, K becomes a connected real graph Γ after we orient each edge.

Next we construct the corresponding band complex Y. For each j = 1, ..., n we do the following. For each each edge of Γ choose maximal (with respect to the inclusion) arcs b so that:

• b ⊂ T_{j};

• ϕ_{j}(int(b)) is disjoint from V ertices(Γ).

We will say that the arcs b correspond to j. Note that the arcs b (corresponding to different indices j) can overlap, there are only finitely many arcs b and they cover the whole graph Γ. The arcs b (and their images in K under ϕ_{j}’s) will be the bases of the union of bands Y. Then for each j = 1, ..., n and each base b corresponding to j, we we attach to Γ a band

b × I = B

where the attaching map on the bottom b is the identity and the attaching map on the top is

ϕ_{j} ∘ δ_{B}

This defines Y. We construct the band complex X as follows:

Step 1. First of all for each j and for each pair of distinct bands B = b × I, B′ = b′ × I (whose bases b = [w, y], b′ = [w, y′] correspond to j) we attach a disk D^{2} ≅ I × I so that:

• The vertical sides of D are identified with the vertical sides w ×I ⊂ B and w ×I ⊂ B′ of the bands B, B′;

• the horizontal edges of D are mapped to the points w and δ_{B}(w) = ϕ_{j}(w) respectively.

Picture!!! Let X′ be the CW complex resulting from this procedure, π_{1}(X′) is isomorphic to the free group on n generators, these generators correspond to the segments

w_{j} × I ⊂ b_{j} × I

where we choose one edge b_{j} and one end-point w_{j} of b_{j} for each pair j.

Step 2. Each relation R_{i} is represented by a vertical loop r_{i} in Y which is based at the vertex x (by the second condition on the finite set S ⊂ T). Then attach to X′ a 2-disk along r_{i}. Repeat this for each i = 1, ..., k.

Note that each subgroup F_{j} of G is generated by closed vertical boundary loops in Y which are based at the vertex z_{j}. Then the subgraph D_{j} in the assertion of Theorem is the union of these loops.

Let X̃ be the universal cover of the band complex X. Cut X open along the top bases dual(b). We get a simply-connected complex X^{0} equal to K with 2-cells and bands B attached along the bottom bases b. We retain the notation X^{0} for a lift of X^{0} to X̃. Then X^{0} is a fundamental domain for the action of G on X̃. There is a canonical mapping

ρ : X^{0} → T

which is the identity embedding K ↪ T on K and is constant on each vertical segment and 2-cell. Since X^{0} ⊂ X̃ is a fundamental domain for the action of G, the map ρ extends to a G-equivariant map ρ : X̃ → T which is a resolution for the G-tree T. We leave it to the reader to verify that the band complex X is regular. □

Remark 14.36 In the above theorem the map ρ : T_{X} → T factors through q : X̃ → T_{X} and becomes a morphism f : T_{X} → T of G-trees. The projection of K ⊂ X^{0} ⊂ X̃ to T_{X} is an isometric embedding as well as the map f : q(K) → K ⊂ T.

We will say that resolving complex X satisfying the property:

for each F_{j} there exists a connected subgraph D_{j} ⊂ ∂Y such that F_{j} is contained in the image in G of π_{1}(D_{j})

is a relative resolving complex with respect to the collection of subgroups F_{i}.

Definition 14.37 A stable G-tree T is called pure if it admits a minimal resolving band complex X such that Y is connected and π_{1}(Y) → π_{1}(X) is onto.

Note that the resolving band complex X constructed in Theorem 14.35 is always such that Y is connected and π_{1}(Y) → π_{1}(X) is onto. However in general the complex X is not minimal.

14.7 Holonomy of vertical paths in X

Pick a band B = b × I in a union of bands Y, let b ⊂ γ, dual(b) ⊂ α where γ, α are edges of Γ. Then B determines an isometry

h_{B} : γ → α

whose restriction to b equals δ_{B} : b → dual(b). If we interchange the top and the bottom of the band B we get the inverse isometry

h_{B}^{−1} : α → γ

We will say that h_{B} is the holonomy of any vertical path in B which connects b to dual(b), we also call h_{B} the holonomy of B. We say that B is orientation-preserving (resp. orientationreversing) if its holonomy is orientation-preserving (resp. orientation-reversing). Suppose that we have a cellular path p : [0, 1] → L where L is a leaf in Y, x = p(0) ∈ b, y = p(1) ∈ c where b, c are bases in Y and a horizontal neighborhood of x in b pushes along p into a horizontal neighborhood of y in c. Then (by subdividing p appropriately) we represent p as a composition p_{1} ∗ ... ∗ p_{n} of paths each of which is contained in a single band B_{j}. Hence we can define the holonomy h_{p} : b → c of p as the composition of holonomies of the paths p_{j}. It is clear that the holonomy of p depends only on the combinatorics of p, i.e. the sequence of bands (B_{1}, ..., B_{n}).

We say that p is orientation-preserving (resp. orientation-reversing) if h_{p} preserves (res. reverses) the orientation. In general this property depends on the choices of orientation on edges β and γ of Γ containing b and c, however if β = γ then it is independent on the orientation of edges of Γ.

If b, c are contained in a single edge β = γ of Γ and p is orientation-preserving we define the translation length ℓ(p) of p as the translation length of the isometry h_{p} : γ → γ.

Here is an interpretation of these notions in terms of actions on trees. Suppose that X is a band complex based on Y which resolves an action of G = π_{1}(X) on a tree T. We assume also that under this resolution each edge γ of Γ embeds isometrically into T and that path p is such that b, c are contained in a single edge γ. Define the closing up p̂ of the path p as the composition λ ∗ p where λ is a path in γ connecting y = p(1) to x = p(0).

Then p̂ determines an element g of G = π_{1}(X, x) which depends only on p. Choice of a lift of x to X̃ defines an isometric embedding ι : γ ↪ T. It is clear that:

• p is orientation-preserving iff the partially defined isometry g : ι(γ) → ι(γ) is orientationpreserving;

• If ℓ(p) is defined then it is equal to the translation length of this partially defined isometry.

In particular, if the translation length of p is different from zero then g is a nontrivial element of G. Moreover, by Lemma 11.7, if ℓ(p) is different from zero then g is an axial isometry of T, the axis of g contains ι(γ) and ℓ_{T} (g) equals ℓ(p).

14.8 An analogue of Kazhdan-Margulis lemma for group actions on trees

It is clear that direct generalization of Kazhdan-Margulis lemma fails for group actions on trees, for instance stabilizer of a point can be arbitrarily large, etc. Nevertheless it turns out that some residual form of Kazhdan-Margulis lemma still holds for stable actions on trees (Theorem 14.41 and Corollary 14.42 in this section).

Definition 14.38 Let J be a nondegenerate subarc of an edge of Γ with a base-point z_{0} in the interior of J. A loop α in X based at z_{0} is said to be J-short if it is a composition p_{1} ∗ λ ∗ p_{2} where p_{1}, p_{2} are paths in the interior of J and λ is a path in a nonsingular leaf of Y.

In this section we assume that X is a band complex which has no faults. In the following two propositions we assume that C(Y) is a minimal component of Y. We let C(Γ) := Γ ∩ C(Y).

Proposition 14.39 For each nondegenerate horizontal segment J ⊂ C(Y) and each z ∈ C(Y) there is ϵ > 0 such that N(z, ϵ) pushes into J.

Proof: Directly follows from minimality of C(Y). □

Proposition 14.40 Let J be a nondegenerate subarc of an edge of Γ with a base-point z_{0} in the interior of J. Then the image of π_{1}(C(Y)) in π_{1}(X) is generated by J-short loops.^{7}

Proof: Recall that Y has only finitely many singular leaves. Let γ be a loop in C(Y) based at z_{0}. Then γ is homotopic to a composition of paths

p_{1} ∗ λ_{1} ∗ p_{2} ∗ λ_{2} ∗ ... ∗ λ_{k} ∗ p_{k}  (41)

where each p_{j} : I → C(Γ) is an injective path in an edge of Γ and each path λ_{i} is contained in a nonsingular leaf. Using the previous proposition subdivide each segment p_{j}(I) into subintervals p_{j}(I_{m}) so that:

• Each p_{j}(I_{m}) pushes onto a (parameterized) segment J_{jm} in J via homotopy H_{jm} : I_{m} × [0, 1] → X, I_{m} = [t_{m}, t_{m+1}];

• the end-points p_{j}(I_{m})^{±} of each p_{j}(I_{m}) belong to nonsingular leaves.

Let σ_{im}^{+}, σ_{im}^{−} be paths in the (nonsingular!) leaves of Y which connect p_{j}(I_{m})^{+} to the initial point of J_{jm} and the terminal point of J_{jm} to p_{j}(I_{m})^{−}:

σ_{im}^{+}(τ) = H_{jm}(t_{m}, τ), τ ∈ [0, 1]

Figure 45: Construction of short loops.

σ_{im}^{−}(τ) = H_{jm}(t_{m}, 1 − τ), τ ∈ [0, 1]

See Figure 45.

Now in the composition of paths (41) we replacee each path p_{j}, j = 2, ..., k − 1, by composition of paths of the form

σ_{jm}^{+} ∗ J_{jm} ∗ σ_{jm}^{−}

where each α_{jm} is contained in a nonsingular leaf. This gives the decomposition of γ as a product of J-short loops. □

Recall that if T is a H-tree then T_{H} is the minimal H-invariant subtree in T.

Theorem 14.41 Suppose that X is a (regular!) band complex resolving a stable G-tree T. Let C(Y) ⊂ Y ⊂ X be a minimal component of Y and let H denote the image of π_{1}(C(Y)) in G = π_{1}(X). If h ∈ H is an element fixing an arc of T then h lies in the kernel of the action of H on T_{H}.

Proof: Choose a subarc J ⊂ C(Y) of an edge of Γ so that J maps via the resolution ρ : X̃ → T to an arc ρ(J) in a maximal stable subtree S ⊂ T_{H}. Let G_{J} be the set of generators of G constructed in Proposition 14.40. For each generator g ∈ G_{J} we have:

g(ρ(J)) ∩ ρ(J) is a nondegenerate arc

Since S is a maximal stable subtree, our discussion in section 14.1 implies that g(S) = S. However T_{H} is minimal with respect to the action of H, hence S = H. Stability implies that if h ∈ H fixes an arc in S it fixes the whole S. Theorem follows. □

7The number of generators is possibly infinite.

Corollary 14.42 Under the assumptions of Theorem 14.41 suppose that α, β are elements of H acting as hyperbolic isometries of T. If the length of I := Axis(α) ∩ Axis(β) is larger than the sum of translation lengths of α and β then Axis(α) = Axis(β).

Proof: Consider the commutator γ = [α, β]. Let J be a subsegment of I adjacent to one of the end-points so that

length(J) = length(I) − ℓ_{T} (α) − ℓ_{T} (β) > 0

Our assumptions imply that the arc J is fixed by the element γ. Hence γ acts trivially on the tree T_{H}. Thus the restrictions of α, β to T_{H} commute. The tree T_{H} contains the axes of α and β, therefore Axis(α) = Axis(β). □

Remark 14.43 This corollary is similar to the behavior of axes of hyperbolic elements g, h in discrete groups of isometries of spaces X with negatively pinched curvature (which follows from the Kazhdan-Margulis lemma): if axes A_{g}, A_{h} of g, h are sufficiently close to each other along subintervals of A_{g}, A_{h} whose lengths are sufficiently long (with respect to the translational lengths of g, h) then A_{g} = A_{h}.

14.9 Moves

In this section we describe a collection of moves that transform one complex of bands X to another complex of bands X′. These moves have the property that:

• π_{1}(X) ≅ π_{1}(X′).

• If X resolves a G-tree T then X′ also resolves T.

• If the resolution X of T was exact then the new resolution X′ is exact as well.

• If we are given a finite collection of subgroups F_{i} ⊂ G as in Theorem 14.35 and X is a relative resolving complex with respect to this collection of subgroups then X′ also is.

Move M0: “Add a 2-cell”. Add a 2-cell D to X along a loop λ in the union

Y ∪ 1-skeleton of X

so that λ meets Y in a union of finitely many vertical fibers and λ is nil-homotopic in X.

Move M1: “Add an annulus”. Let J be a subarc of an edge of Γ. Attach to X a weight zero band B with base J and a 2-cell D along the loop that is formed by one of the vertical fibers of B.

Move M2: “Subdivide a band”. Let B = [p, q] × I be a band of Y. Pick a point x ∈ (p, q). We obtain Y′ by replacing B with the two bands B_{1} = [p, x]×I and B_{2} = [x, q]×I by splitting B open along the vertical segment x × I. As we result we get two copies of is vertical segment σ, σ′. Attach a 2-cell D along the loop σ ∪ σ′. The cell D is called a subdivision cell.

Move M3: “Split a point”. Suppose that z is a point of Γ that does not meet the interior of any base, however z is a local cut-point and belongs to at least one base. Split Γ open at z, the resulting graph Γ′ has a collection of vertices Z := {z_{1}, ..., z_{s}} corresponding to z. Add to Γ′ a cone Cone(Z) over Z (this cone will be a part of X′ − Y′). Attach bands and cells to Γ′ ∪ Cone(Z) as before to get X′.

Lemma 14.44 By applying a finite sequence of the moves M2 and M3 to a band complex X we may arrange that X has no faults.

Proof: We recall that according to our discussion in Section 14.3 the band complex X has only finitely many faults in X, each of which is a finite subgraph in a singular leaf, the frontier of this subgraph consists of critical points. We split Y open along each fault using the moves M2 and M3. Recall that there are two types of critical points on the singular leaves of Y (local cut points and dead ends), the Move M3 eliminates the local cut-points, the Move M2 eliminates dead ends. These moves also do not introduce new critical points (on frontiers of faults). Thus using Moves M2 and M3 we eliminate all critical points (on frontiers of faults), as the result we get a band complex without faults. □

We can assume that the boundary of each 2-cell in X has a finite subdivision so that if the interior of an edge of the subdivision meets Y then this edge is contained in Y.

Theorem 14.45 Let T be a stable G-tree for a finitely-presentable group G. Suppose that the band complex X has no faults and resolves T. Then G has a graph of groups decomposition such that:

• The action on T of each vertex group of the decomposition is either trivial or pure.

• The action on T of eaach edge group is trivial. Moreover, each such group is contained in the image in G of the fundamental group of a leaf of X.

• If we are given a finite collection of subgroups F_{i} ⊂ G as in Theorem 14.35 and X is a relative resolving complex with respect to this collection of subgroups then each F_{i} is contained in the image of one of the vertex groups of the decomposition of G.

Proof: By Theorem 14.26, each component C(Y) of Y is either minimal or simplicial, in the latter case π_{1}(C(Y)) fixes a point in T.

Note that cl(X − Y) has a natural CW-complex structure. We give X the structure of a graph of spaces as follows:

(a) The vertex spaces are the components of Y and of cl(X − Y).

(b) The edge spaces are the components of intersection of cl(X − Y) and Y. Note that each edge space is contained in a leaf of Y.

Remark 14.46 This decomposition of X does not satisfy one of the properties of a graph of spaces: the images of the edge-spaces need not be pairwise disjoint. To resolve this problem we use (as vertex-spaces) instead of the components of X −Y their deformation retracts and (as edge-spaces) instead of the components Σ of the intersection Y ∩ cl(X − Y) appropriate neighborhoods of Σ in X which admit deformation retractions to Σ. The details of this are left to the reader.

If vertex space X_{v} is a simplicial component of Y or a component of cl(X − Y) then the action of π_{1}(X_{v}) on T has a global fixed point. Otherwise X_{v} is a minimal component of Y. Hence in this case the action of π_{1}(X_{v}) on T is pure. □

Note that vertex and edge groups in this graph of groups decomposition of G can be non-isomorphic to the fundamental groups of vertex and edge spaces, they are merely the quotients of such groups. Hence, apriori vertex and edge groups can fail to be finitely presentable. (We choose the coverings X̃_{v} → X_{v} using the restrictions of the universal covering X̃ → X to the inverse images of X_{v} in X̃.) At this stage we also cannot control the group-theoretic structure of the edge groups except for the edge groups incident to the vertices corresponding to simplicial components of Y.

Move M4: “Slide”. Suppose that we have a pair of distinct bands B = b × I and C = c × I in a band complex X. Assume that f_{b}(b) ⊃ f_{c}(c). We change the union of bands Y to Y′ by changing the attaching map f_{c} to

f′_{c} := f_{dual(b)} ∘ δ_{B} ∘ f_{b}^{−1} ∘ f_{c}

Informally, we “slide” the base c from b to dual(b) along the band B so that each point of c “slides” along a vertical segment of B, as the result the band C in X transforms to a band C′ in X′, see Figure 46.

Figure 46: Slide.

We say that b is the carrier of the slide and c is the carried base.

Remark 14.47 Note that the move M4 can possibly increase the complexity. Here are the possibilities how it may happen:

(1) The carried base c has weight zero.

(2) The carried base c has weight 1/2.

(3) The carried base c has weight 1 and is contained in a block β of zero complexity, which means that β contains only the bases b and c whose weights are equal to 1. See Figure 47. (The complexity will not change if weight(b) = 1/2.)

To avoid the increase of complexity in the first two cases we should slide along B both bases of the band C in a single move.

Figure 47: .

Lemma 14.48 Suppose that we have a nondegenerate segment s ⊂ Γ ⊂ X such that s pushes onto a segment s′ ⊂ Γ. Then there is a finite sequence of moves M1 and M4 which transforms X to a band complex X′ whose underlying union of bands Y′ equals Y ∪A′ where A′ is a band with the bases s, s′.

Proof: First we attach a band A of zero weight to s via the move M1. Now keep one of the bases a of A attached to s and realize the pushing of s onto s′ by a sequence of slides of a′ = dual(a) that terminates when we slide a′ onto s′. □

The following two lemmas are used to describe the next (and the last) Move M5 in presence of weight 1/2 bases.

Lemma 14.49 Let B = b × I be a band with weight(b) = 1/2. Then there is a sequence of moves that replaces B by a pair of bands B′ = b′ × I and A = a × I, where weight(b′) = 1, weight(a) = 0 and b′ = a (i.e. the images of these bases in Y coincide), b′ ∩ dual(b′) is a single point.

Proof: Subdivide B along the midline into two weight 1 bands (Move M2), then slide one of these bands over the other one (Move M4). □

Lemma 14.50 Let B = b × I be a band with weight(b) = 1/2, b = [u, v], let m denote the midpoint of b. Pick a point x ∈ (m, v) and denote by y ∈ (u, m) the point dual to x. Then there is a sequence of moves which replaces B by weight 1/2 band B′ = b′ × I, b′ = [x, y], a weight zero band D = d × I with d = [u, y] and a weight 1 band E = e × I with e = [x, v] and dual(e) = d.

Proof: First subdivide along the vertical loop in B that contains x, y. Then slide the band D along the band E (notice that they share the common base e). See Figure 48. □

Figure 48: Subdivision of a Moebius band.

Definition 14.51 A subarc c of a base b is free if either:

1. b has weight 1 and the interior of c does not meet any other base of positive weight.

2. b has weight 1/2, the interior of c does not contain the midpoint of b and the bands b, dual(b) are the only positive weight bases that meet the interior of c.

Below we describe the Move M5, I recommend the reader first to assume that the free arc c of a base b does not meet interiors of any bases (except b) and that weight(b) = 1. See Figure 50.

Move M5: “Collapse from a free subarc”. Let B = b × I be a band in Y and c = [p, q] be a free subarc of b.

(1) We first use the Move M2 to subdivide all zero weight bands meeting c until for every zero weight base d that meets interior of c we have: d ⊂ c. This step creates subdivision disks. Let X_{1} be the resulting complex of bands.

(2) Next, use Move M4 to slide over B (into dual(c)) all weight zero bases d contained in c. As the result the subdivision disks become annuli (called subdivision annuli).^{8} Now c does not meet interior of any base but b (except if weight(b) = 1/2 then c also meets dual(b)). Let X_{2} denote the resulting complex of bands.

Figure 49: Creation of the subdivision annuli.

(3a) If weight(b) = 1 then collapse c × I to dual(c) ∪ ∂c × I. In the special case when one of the end-points x of c is also an end-point of b and the open segment x × (0, 1) does not meet any cells in X_{2}, we eliminate x × (0, 1). We modify the graph Γ by removing the interior of c. The result is the complex of bands X′.

There are two other cases when weight(b) = 1/2 and c is contained in one half of b.

(3b) Suppose that weight(b) = 1/2 and one of the end-points x of c is the mid-point of b. Then apply Lemma 14.49 to the band B transforming it to an annulus A and a weight 1 band B′ so that the interior of c is disjoint from A. Now c is a free arc in a weight 1 base of B′ and we can apply (3a).

8Here is a description how these annuli look like. On the Step (1) we split open a weight zero band A into two bands, say A′, A′′ where we suppose for simplicity that A′ is disjoint from the interior of c and the base of A′′ is contained in c. As the result we get two vertical circles: one in A′, the second is in A′′. These circles are identified in X and homotopic in X_{1} and share a common end-point x of c, the homotopy is given by a subdivision disk. After the sliding (Move M4) the circles become disjoint, however they still have to be homotopic: the homotopy is given by a subdivision annulus that contains the vertical interval x × I of B.

See Figure 49.

(3c) Suppose that (3b) does not occur, let x be the end-point of c which is the closest to the midpoint m of b (and m ≠ x). Then similarly to the case (3b) apply Lemma 14.50 and reduce the problem to the case when c is a free arc in a weight 1 base.

This finishes the description of the Move M5.

Remark 14.52 (1) Suppose that p is the end-point of c that belongs to the interior of b. Then the open vertical segment p × (0, 1) ⊂ X′ meets cl(X′ − Y′) only along the subdivision annuli.

(2) If D_{j} ⊂ ∂Y is a connected subgraph which carries the subgroup F_{j} and D′_{j} is the corresponding subgarph in ∂Y′ then the intersection D_{j} ∩ p × (0, 1) is empty. See Figure 50.

Figure 50: Collapse in the Move M5.

14.10 Preliminary motions of the Rips machine

We start with action of a finitely-presentable group G on a tree T, in the relative case we are given a finite collection of finitely generated subgroups F_{i} each of which fixes a point in T. Then we construct a band complex X which resolves this action, the map ρ : X̃ → T from the universal cover of X to T is as in Theorem 14.35.

We now use Moves M2 and M3 to transform X to another band complex (which we still denote X) so that the new band complex X satisfies the following properties:

(A1) The real graph Γ is the disjoint union of its edges. Each edge is a block.

(A2) The union of bands is a disjoint union of components each of which is either minimal or simplicial and each simplicial component is a trivial I-bundle.

(A3) If B = b × I and A = a × I are weight 1/2 bands such that a, b share midpoints then B = C (this is achieved via the Move M2).

(A4) The resolution ρ : X̃ → T embeds lifts of bases into T, where X̃ is the universal cover of X.

(A5) The complex X has no faults.

14.11 The Machine

Definition 14.53 (A long band.) A sequence B_{1}, ..., B_{n} of weight 1 bands forms a long band B provided that:

1. The top of B_{j} is contained in the bottom of B_{j+1} for j = 1, 2, ..., n − 1.

2. B_{i} ∩ B_{j} = ∅ if |i − j| > 1.

3. B_{1} ∪ .... ∪ B_{n} intersects other bands of positive weight in a subset of the union of the bottom b of B_{1} and the top b′ of B_{n}.

See Figure 51.

If J = J_{1} is an arc on the bottom of B_{1} then we get a collection of arcs J_{2}, ..., J_{n} on the bottoms of B_{2}, ..., B_{n} so that

J_{i+1} = δ_{B_{i}}J_{i}

We will refer to the bottom of B_{1} and the top of B_{n} are the bottom and top of the long band B. We let δ be the composition δ_{B_{n}} ∘ ... ∘ δ_{B_{1}}.

Figure 51: A long band.

Suppose that b = [x, y] is a base of weight 1/2 with the midpoint m. Then the halves of b are the intervals [x, m] and [m, y]. We will refer to these halves as half-bases.

Definition 14.54 (Isolated bases.) A weight 1 base is isolated if its interior does not meet any other base of positive weight. A half h of a base b (of the weight 1/2) is isolated if the interior of h meets no positive weight bases other than b, dual(b).

A weight 1 base b is semi-isolated if a punctured neighborhood in b of one of its endpoints p meets no other positive weight bases. A half h of a weight 1/2 base b is semiisolated if a punctured neighborhood of an end-point p of h meets no positive weight bases other than b, dual(b). In these cases we say that b and h are semi-isolated at p.

Remark 14.55 Note that if there is isolated half then we get contradiction with the minimality assumption, the same is in the case when we have semi-isolated half which is semiisolated at the mid-point p of the corresponding base. The pushing-saturated non-dense subset is the mid-loop of the Moebius band through p.

Now we will give the list of rules (R0)–(R4) which describe the Process I as applied to a minimal component C(Y) of a union of bands Y (satisfying properties (A1)–(A5)), the Process I transforms X to a band complex X′:

(R0) If there is no free subarc of a base b of C(Y), then we go to the Process II. Otherwise we find a maximal (with respect to inclusion) free subarc J of a base b of C(Y). Use the Move M5 to collapse from J. It could be that there are several such subarcs J, here is how we choose one of them:

(R1) If there is an isolated (half-) base c take J = c (if there are several choices pick at random).

(R2) Suppose there are no isolated (half)-bases but there is a semi-isolated (half- ) base c which is semi-isolated at end-point p. Then choose a maximal free subarc J ⊂ c that contains p.

(R3) If there are no isolated or semi-isolated (half-) bases then choose J at random. (R4) Treat the “long bands” as units. Namely, if we have a long band (B_{1}, ..., B_{n}) and

we use J in the bottom of B_{1} then we perform n consecutive collapses, i.e. we collapse from J then from the bottom of B_{2}, etc, and finally from the bottom of B_{n}. These n operations are treated as a single move producing X′. We let b′ denote the top of the band B_{n}.

Remark 14.56 Let J be a free arc in the bottom of a long band B and δ(J) is defined as in Definition 14.53. Then:

• The arc δ(J) contains no free subarcs. Indeed, otherwise we would have vertical leaves in B terminating at δ(J) which are not dense in C(Y), this contradicts minimality of C(Y).

• If p ∈ intδ(J) is an end-point of base, then there is a base c distinct from the top of B which contains p. We verify this is the case when top of B_{i} is the bottom of B_{i+1} for all i. Connect p by a vertical segment λ to a point q = δ(p) ∈ J. If the top of B is the only base containing p then the point p is a “dead end” and the segment λ is a fault in the leaf through p which is a contradiction.

Note that the Process I does not change the number of components of Y.

Lemma 14.57 Suppose that X′ is produced from X by Process I. Then Complexity(X′) ≤ Complexity(X).

Proof: We will consider the “generic” case when (R3) is applied, the rest is left as an exercise to the reader. Let B be the (long) band whose bottom base b contains J and γ be the block containing b. Then after collapse from J the block γ splits into two or three components depending on the location of δ_{B}(J). We consider the case when γ splits into two components γ′, γ′′ of nonzero complexity and δ_{B}(J) is contained in a block α ≠ γ so that Complexity(α) is positive. Let α′ be obtained from α as the result of collapse from J (notice that α′ is still a single block by the Remark 14.56). Then after collapse from J we have:

Complexity(γ′) + Complexity(γ′′) = −3 + ∑_{c is a base in γ} w(c) = Complexity(γ) − 1

Complexity(α′) ≤ Complexity(α) + 1

Therefore

Complexity(γ′) + Complexity(γ′′) + Complexity(α′) ≤ Complexity(γ) + Complexity(α)

As far as other blocks are concerned, their complexity does not change.

Remark 14.58 There is a special case when B is a long band and the Move M5 splits each intermediate base b_{i}, 1 < i < n, into two components. However each of these b_{i} forms a block of zero complexity and the splitting into two pieces does not change this fact.

This concludes the proof. □

The following proposition describes what happens as the result of application of the Process I.

Proposition 14.59 Suppose that we have an infinite sequence X_{0}, X_{i}, X_{2}... of band complexes so that X_{i+1} is obtained from X_{i} via the Process I for each i > 0. Then the following hold:

1. The choice of J as in (R1) occurs only finitely many times.

2. The number of weight 1/2 bases does not increase.

3. There is a number N so that all bands in each X_{i} can be organized into at most N long bands.

4. For any n there is a segment of length n contained in some leaf so that this segment is eventually collapsed.

5. The choice of J as in (R3) occurs infinitely many times.

Proof: (1) Collapse from an isolated base J will eliminate one block and decrease complexity of the block α containing the base dual(J) unless α contains only two bases one of which contains dual(J) in which case B is a part of a “long band” and the rule (R4) is applied. Thus in any case the complexity of C(Y) will decrease.

(2) Obvious.

(3) Using the previous lemma we may assume that complexity of X_{i} is a constant σ for all i. Suppose that the number of long bands of X_{i+1} is larger than the number of long bands in X_{i}. This can happen only if the rule (R3) is applied, i.e. there are no (semi-) isolated bases and half-bases. Hence each long band contains a base in a block of positive complexity, therefore the number of long bands in X_{i} is bounded by 3σ and the number of long bands can increase by at most 1 in X_{i+1}. We conclude that in each X_{j} the number of long bands is at most

3σ + number of long bands in X_{0}

(4) Define inductively the functions N(k, x), N(k), k ∈ ℕ where x belongs to a free subarc J(x) of X_{N(k)} that is eventually collapsed under the Process I.

Remark 14.60 Note that it is apriori possible that certain free arc in a base of X_{0} is never collapsed.

Set N(0, x) := 0, N(0) := 0. Suppose that N(k, x), N(k) are defined. Let N(k +1, x) be the least integer such that the arc J(x) containing x is collapsed at the stage N(k + 1, x). Note that for fixed k the function N(k + 1, x) is constant on the interior of each free arc in X_{N(k)}. Let

N(k + 1) := max_{x} N(k + 1, x).

Clearly lim_{k→∞} N(k) = ∞. Each point x ∈ X_{0} that is collapsed at the stage N(n) belongs to a vertical segment of length n in X_{0} that is totally collapsed after the stage N(n) of the Process I.

(5) Recall that η(X_{i}) (the height) is the total number of bands in X_{i} (here we do not count long bands as a single band). Suppose that only collapses of the type (R2) occur. This implies that the height η(X_{i}) = η does not depend on i. Our goal is to show that these assumptions imply that Process I will terminate after finitely many steps.

By (4) for each n > 0 there is an (oriented) vertical segment λ of length n which is collapsed after the stage N(n) of the Process I. Choose n > 4η. Then the segment λ intersects a band B = b × I in X in at least 5 distinct vertical segments λ_{j}. Among these 5 segments we choose three segments λ_{1}, λ_{2}, λ_{3} which have the following property:

Orient λ_{j} using orientation of λ and order these segments so that λ_{1} is the first and λ_{3} is the last with respect to the linear order on λ. Let x_{j} denote the first point of λ_{j} that belongs to b. Then we require the paths s_{1} = [x_{1}, x_{2}] and s_{2} = [x_{2}, x_{3}] to be orientation-preserving (see Section 14.7).

Then the segments λ_{j} are separated by the distance θ > 0 which does not depend on B and λ but depends only on the original complex of bands X.^{9}

At least one of the segments (say λ_{2}) is separated from the vertical boundary of B by two other segments λ_{1}, λ_{3}. Hence the distance from λ_{2} to the both vertical boundary segments of B is at least θ. By our assumption the segment λ is collapsed after the stage N(n) of the Process I and only the rule (R2) is applied, thus after the stage N(n) the length of at least one base (namely b) had decreased by at least θ. We apply this argument inductively to conclude that between the stages N(kn), N((k + 1)n) the length of at least one base had decreased by at least θ. This implies that eventually all bases will be eliminated and the Process I will terminate after finitely many stages. This is a contradiction. □

Now we descibe the Process II. Let C(Y) be a minimal component of Y. We suppose that the Process I does not apply to C(Y). Hence:

(*) The weight of each point z ∈ C(Γ) := Γ ∩ C(Y) is at least 2. (Otherwise z would be contained in a free arc.) We identify components of C(Γ) with disjoint closed intervals in ℝ. This gives C(Γ) a linear order. Let F be the first component of C(Γ) and x be the initial point of F. Let b denote the longest base of positive weight containing x (take b to have weight 1 if such base exists, if there are several such bases then we choose one at random).

Operation 1. Slide from b all those positive weight bases c contained in b (except of b, dual(b)) whose midpoint is moved away from x as the result of the slide. The base b is the carrier of each of these slides. We will say that the base c was carried during this operation.

Operation 2. Collapse from the maximal free initial segment J of b. Note that the segment J is necessarily nondegenerate.

Definition 14.61 The bands b × I and c × I are said to have participated in the Process II.

The Process II consists of consecutive application of Operation I and Operation II. Let X′ be the result of application of the Process II. We adopt the following convention

about labelling the bases of positive weight in X and X′:

(i) If c ⊂ b is a base which participated in the Process II, then we retain the name c for the base in X′ obtained from c via sliding along B.

(ii) We retain the name b for the base in X′ obtained from b by collapsing the segment J.

9The holonomy transformations g_{1}, g_{2} ∈ G determined by the segments s_{1}, s_{2} belong to a finite subset of G. The number θ is the minimal translation length of these transformations.

With this convention it makes sense to talk about dynamics of bases of X under the a sequence of iterations of the Process II, for instance we talk about the limiting position and the limiting length of a base d of X.

Proposition 14.62 If X′ results from X via the Process II then:

1. Complexity(X′) ≤ Complexity(X).

2. If Complexity(X′) = Complexity(X) then (*) holds for X′.

Proof: (1) According to the Remark 14.47, the slides in the Operation 1 do not increase the complexity, except possibly the last slide where we slide from a block of zero complexity (case (3) in Remark 14.47). Let β denote the block in X containing b. If the last slide from b increases complexity then after all these slides only the base b of the block β remains and Operation 2 eliminates the block β completely thus decreasing the complexity.^{10} The Operation 2 clearly can not increase the complexity.

(2) Since (*) is satisfied by X there are the following reasons it might fail for X′:

(a) There is a base c that after sliding along B becomes a base c′ of weight 0 (this decreases the total weight of bases containing points of int(c′)). However in this case Complexity(X′) < Complexity(X).

(b) Similarly a base c might become a base of weight 1/2, this again leads to decrease of the complexity by 1/2.

(c) Let J denote the maximal initial free segment of b that is collapsed by the Operation II. It might happen that w(z) ≤ 2.5 for some z ∈ δ_{B}(J) after the Operation I and the Operation II reduces the weight of z by 1. However this would imply that the base b of X contains a free subarc, in which case we have to apply the Process I instead of the Process II to the complex X. □

If the complexity decreases, we terminate the Process II and continue with the Process I.

Proposition 14.63 Consider a sequence X_{0}, X_{1}, ... of band complexes so that for each i ≥ 0 the complex X_{i+1} is obtained from X_{i} via Process II. Then one of the following holds:

1. The sequence terminates after finitely many steps.

2. (The surface case.) For all sufficiently large i and each point of Γ_{i} = Γ(X_{i}), which is not an end-point of a base, we have: weight of z equals 2.

3. (The toral case) (a) Every base has a limiting position in Γ. (b) Some base b is a carrier and is carried infinitely often. (c) The length of the base b does not converge to zero. (d) The bases b, dual(b) have the same limiting positions in ℝ.

Proof: Suppose that neither (1) nor (2) occur, thus we have the toral case. Then the complexity of this sequence of band complexes is constant.

(a) Every base b has a limiting position since on each step we have: either b is carried, thus it moves to the right and its length remains constant, or b is the carrier, in which case the right end-point of b does not move and the length of b does not increase. This Proves (a). Without loss of generality we may assume that:

(i) If a band participates in the Process II it does so infinitely often. (ii) Weights of bases do not change as the result of the Process II.

10There is a special case when β has only two bases. These bases must be equal since X has no free arcs.

We leave this case to the reader.

Remark 14.64 It is apriori possible that during the Operation I we have a band c × I of weight 1 which is carried and after that becomes a band of weight zero or 1/2 (an annulus or a Moebius band). However this would lead to decrease of complexity which we assume is not the case.

Recall that we identify Γ with a disjoint union of intervals in ℝ. Let D_{i} denote the total measure of the subset of points in Γ_{i} between the initial point and the minimal point of Γ that lies in a band that never participates. Define

τ_{i} := ∑{mes(b) · weight(b)|b is a base of X_{i} which eventually participates}
excess(X_{i}) := τ_{i} − 2D_{i}

Lemma 14.65 We have: excess(X_{i}) > 0 and excess(X_{i}) = excess(X_{i+1}).

Proof: The first assertion follows from the fact that the weight of each point is at least 2 and that the “surface” case (2) does not occur.

To verify the second assertion we analyse the Operations II and I. As the result of the Operation II (when we collapse the maximal free initial subinterval J contained in the carrier d) both 2D_{i} and τ_{i} decrease by 2 · mes(J) · weight(d), hence the excess does not change. During the Operation I we only slide bands over each other and this does not change their weights (see the Assumption (ii) above), hence τ_{i} and D_{i} do not change as well. □

It follows from the above lemma that there is a base b of positive weight that participates infinitely often and its length does not converge to zero. We claim that the base b has to be carried infinitely often. Suppose not. Then the base dual(b) is also carried only finitely many times. Thus without loss of generality we can assume that each time the base b (and dual(b)) participates in the Process II it is the carrier. Under these assumptions the right end-points of b and dual(b) and the translation length ν of b (if b is orientation-preserving) remain (eventually) constant for all members of the sequence X_{i}. There are several cases to consider:

(i) b and dual(b) have disjoint interiors and the corresponding band B = b × I is orientation preserving. Then each base can be carried by b only finitely many times, hence the Process II eventually terminates, which is a contradiction.

(ii) Suppose that B is orientation-reversing. Then a base c can be carried by b at most once and we get the same conclusion as in (i).

(iii) Suppose that the band B has weight 1 and is orientation-preserving. Then each base c which is carried from b moves away (to the right) from its initial point by the translation length ν > 0 of b. If b is a carrier infinitely many times then some base c moves to the right by ν infinitely many times which is impossible.

This proves (b) and (c). Finally we consider (d). If the base b under consideration has the weight 1/2 there is nothing to prove. Suppose weight(b) = 1 and q > 0 is the distance between the mid-points of b, dual(b). Each time the base b is the carrier, a base carried by b moves the distance q to the right. Hence the distance q must tend to zero as we proceed with the Process II. □

14.12 Machine Output

Recall that we start with a finitely presentable group G acting on a tree T and a finite collection of finitely generated subgroups F_{j} ⊂ G each of which fixes a point in T. We construct a band complex X which resolves an action of G = π_{1}(X) on tree T (relative to the collection of subgroups F_{j}) as in Theorem 14.35.

Definition 14.66 If eventually only the Process I is applied to C(Y) then C(Y) is said to have the thin type^{11}. In the case when eventually only the Process II is applied and excess = 0 then we say that C(Y) has the surface type. Otherwise the Process II is applied and excess is a positive constant in which case C(Y) has the toral type.

We let \overline{Y} denote the union of bands obtained from Y by omitting all zero weight bands. Given a component C(Y) of Y we shall use the following notations:

Set \overline{C(Y)} := C(Y) ∩ \overline{Y}. We let ι denote the homomorphism ι : π_{1}(C(Y)) → G induced by the inclusion, H := ι(π_{1}(C(Y))), let N be the normal subgroup of H normally generated by the images of the fundamental groups of annuli in the closure of C(Y) − \overline{C(Y)} and let \overline{H} := H/N.

Now we start case-by-case consideration of the Machine output.

(i) Output of the Process II: the surface type case.

Recall that a group is called cyclic if it is 1-generated, this includes the trivial group, ℤ/n and ℤ.

Proposition 14.67 Suppose that C(Y) is a component of Y which has the surface type. Then we have:

1. For each leaf L of C(Y) the image of π_{1}(L) in G = π_{1}(X) admits a map onto a cyclic group Δ so that the kernel is contained in the kernel of the action of H on T_{H}.

2. Suppose that X is pure (thus C(Y) = Y, G is the image of π_{1}(Y) in π_{1}(X)) and has the surface type, then there is a short exact sequence:

1 → Ker(G ↷ T) → G → π_{1}(Ø) → 1

where Ø is a cone type 2-dimensional orbifold. Each of the subgroups F_{j} of G maps into either peripheral or torsion subgroup of π_{1}(Ø).

Proof: Without loss of generality we can assume that only Process II can be applied to C(Y) and excess(C(Y)) = 0. If C(Y) has any bases of the weight 1/2 then we apply Lemma 14.49 to the corresponding Moebius bands and transform each of these bases to a pair of bases of weight 1 (with disjoint interiors) and a pair of equal bases of zero weight (corresponding to an annulus). This way we eliminate all Moebius bands and still retain w(z) = 2 for each point z ∈ C(Γ) that is not an end-point of a base. Now \overline{C(Y)} is obtained by attaching bands to a collection of intervals in ℝ so that: each point is either an end-point of a base (recall that this can not be a local cut-point) or it belongs to exactly two bases. This implies that \overline{C(Y)} is a surface (with boundary) S. Each nonsingular leaf of \overline{C(Y)} is homeomorphic to ℝ. Recall that no singular leaf contains a proper compact pushing-saturated subset, hence each singular leaf of \overline{C(Y)} is a boundary loop of S with finite (necessarily nonzero) number of rays attached. Hence each singular leaf in \overline{C(Y)} has infinite cyclic fundamental group. Let L be a closed vertical loop in an annular band in C(Y). Then the cyclic subgroup Δ of G (which is image of π_{1}(L)) has a fixed arc on the tree T, hence Δ lies in the kernel of the action of H on the tree T_{H} (see Theorem 14.41). In general the fundamental group of a leaf L of C(Y) has a normal subgroup N ∩ π_{1}(L) (which is contained in the kernel of the action on T_{H}), the quotient group is cyclic (it corresponds to L ∩ \overline{C(Y)}). This proves the first assertion of the proposition. Now consider the second assertion. We have a short exact sequence

1 → N → H → \overline{H} → 1

11In this case the widths of bands in C(Y) converge to zero, thus the name.

Clearly N = Ker(G ↷ T). The group \overline{H} is the fundamental group of \overline{Y} with 2-cells attached along boundary components, hence \overline{H} is isomorphic to the fundamental group of a cone type 2-dimensional orbifold Ø. This proves the Proposition in the absolute case (when we are not given a collection of finitely generated subgroups F_{i} fixing points in T). Each of the subgroups F_{i} is conjugate into the fundamental group of one of the “boundary leaves” of Y (see Theorem 14.35). Thus F_{i} projects either to a peripheral or to a torsion subgroup of \overline{H}. □

(ii) Output of the Process II: the toral case.

We suppose that C(Y) is a toral component of Y ⊂ X and X = X_{1}, X_{2}, ... is a sequence of band complexes which is obtained from X via Process II applied to C(Y). By Proposition 14.63 we may assume that for each i the component C(Y_{i}) has only one block. Let J denote the block of C(Y). Each band of Y_{i} determines a partial isometry of J, extend this isometry to the isometry of ℝ and let B(Y_{i}) ⊂ Isom(ℝ) denote the group generated by these isometries. Clearly this group is finitely generated and it is easy to see that B = B(Y) = B(Y_{i}) for all i. The group B is called the Bass group of C(Y). The elements of B are either translations or symmetries.

Lemma 14.68 Let B ⊂ Isom(ℝ) be a finitely generated subgroup.

1. Suppose that we have a sequence of translations τ_{j} : x ↦ x + t_{j}, j = 1, 2, ..., in B −{1} such that lim_{j} t_{j} = 0. Then this sequence contains translations τ_{i}, τ_{j} such that t_{i}/t_{j} ∉ ℚ.

2. Suppose that we have a sequence of distinct symmetries σ_{1}, σ_{2}, ... in B such that lim_{j} σ_{j} = σ_{0}. Then this sequence contains elements σ_{i}, σ_{j}, σ_{k} such that the ratio of the translation lengths of σ_{i}σ_{j} and σ_{i}σ_{k} is irrational.

Proof: We prove the first assertion of Lemma, the second is similar. Consider the subgroup B′ of B generated by τ_{1}, τ_{2}, .... Since B is finitely generated virtually abelian group, the subgroup B′ is also finitely generated. Moreover, there is a number N so that the elements τ_{1}, ..., τ_{N} generate B′. The subgroup B′ ⊂ Isom(ℝ) is not discrete. Hence there are two generators τ_{i}, τ_{j} of B′ such that t_{i}/t_{j} is irrational. □

Lemma 14.69 We can apply moves M0–M4 and transform X to a band complex X′ so that:

There are two bands A = a × I, A′ = a × I in X′ which are orientation–preserving, they determine partial isometries α, α′ of J so that the domain of the composition αα′α^{−1}(α′)^{−1} is a nondegenerate segment and the ratio of the translation lengths ℓ(α)/ℓ(α′) is irrational.

Proof: Recall that according to Proposition 14.63 there is a base b in C(Y) which is a carrier and is carried infinitely often in the sequence X_{j}. Let b_{i} denote the base in X_{i} corresponding to b. The bases b_{i} are convergent to a certain nondegenerate segment. By the same Proposition 14.63 we can choose a subsequence in X_{i} so that the sequence of isometries h(b_{i}) ∈ B of J determined by b_{i} is convergent. (Recall that B is the Bass group). The limit is the identity in the orientation-preserving case. In particular, all b_{i} are either orientation-preserving or reversing. Now we apply Lemma 14.68 to the sequence h(b_{i}) ∈ B.

We first consider the orientation-preserving case. Then there are bases b_{i}, b_{j} so that the ratio of the translation lengths of h(b_{i}), h(b_{j}) is irrational. We use Lemma 14.48 to attach a band A = b_{i} ×I to b_{i}, dual(b_{i}) and a band A′ = b_{j} ×I to b_{j}, dual(b_{j}). The partial isometries α, α′ are obtained by restriction of h(b_{i}), h(b_{j}) to b_{i}, b_{j}. Recall that

lim_{i} b_{i} = lim_{j} b_{j}, lim_{i} h(b_{i}) = lim_{j} h(b_{j}) = id

Hence, if i, j are sufficiently large, the composition of partial isometries αα′α^{−1}(α′)^{−1} has nondegenerate domain. This concludes the proof in the orientation-preserving case.

Suppose now that all the bases b_{i} are orientation-reversing. Then similarly to the above argument we find b_{i}, b_{j}, b_{k} (i < j < k) so that the ratio of the translation length of h(b_{i})h(b_{j}) and h(b_{i})h(b_{k}) is irrational. Recall that the sequence of bases b_{s} is convergent to a certain nondegererate segment in J. Therefore we can choose i so large that b_{ij} := b_{i}∩b_{j}, b_{ik} := b_{i}∩b_{k} are nondenenerate segments. Consider the sub-bands B_{ij} = b_{ij} × I ⊂ b_{j} × I = B_{j} and B_{ik} = b_{ik} × I ⊂ b_{k} × I = B_{k}. We subdivide the bands B_{j}, B_{k} (Move M2) so that the bands B_{ij}, B_{ik} become bands in the new band complex X′. Now we slide the bases b_{ij}, b_{ik} from b_{i} (Move M4). This produces bands B′_{ij}, B′_{ik} such that translation lengths of h(B′_{ij}), h(B′_{ik}) are equal to the translation length of h(b_{i})h(b_{j}) and h(b_{i})h(b_{k}). We let A := B′_{ij}, A′ := B′_{ik}. To get nondegenerate domain for the composition αα′α^{−1}(α′)^{−1} we again choose sufficiently large i. □

Proposition 14.70 (1) The group H has an invariant line in T and H fits into a short exact sequence

1 → Ker(H ↷ T_{H}) → H → A → 1

where A contains a free Abelian subgroup of the index 2. (2) Let λ be a leaf in C(Y). Then the image π of π_{1}(λ) in H fits into a short exact sequence

1 → K → π → Φ → 1

where Φ is a group of order ≤ 2 and K fixes an arc in T.

Proof: We first prove (1). Let A, A′ be the bands constructed in the previous lemma. Let α, α′ denote the elements of H corresponding to A, A′; let ρ(J) be the image of J in T so that α, α′ are partial orientation-preserving isometries of ρ(J). Then the commutator [α, α′] fixes a nondegenerate subarc I ⊂ ρ(J). Therefore, by Theorem 14.41, the commutator [α, α′] fixes T_{H} pointwise. We conclude that α, α′ act as translations on T_{H} with the same axis L. By Lemma 14.69, the ratio of translational lengths of α, α′ is irrational. Therefore the subgroup F generated by α, α′ contains nontrivial translations with arbitrarily small translational lengths. By Proposition 14.40 the group H is generated by J-short elements, let γ be one of the generators. We will show that L is invariant under γ. Let s ⊂ ρ(I) denote a closed subinterval such that s′ := γ(s) ⊂ ρ(J).

Case I: the partial isometry of J determined by γ is orientation-preserving. By Lemma 11.7 the isometry γ : T → T is axial and its axis contains s ∪ s′. Let ϵ denote the diameter of the subset s ∪ s′ in ρ(J). Choose an element w in F − {1} whose translational length is so small that ℓ(w) + ℓ(γ) < ϵ. Then Corollary 14.42 implies that the axes of γ and w coincide which concludes the proof in the Case I.

Case II: the partial isometry of J determined by γ is orientation-reversing. Let x, x′ denote the mid-points of s, s′. Let t′ denote the subinterval of s′ centered at x′ whose length is the half of the length of s. We again choose w ∈ F − {1} whose translational length is less than the length of t′. Then w′ := γwγ^{−1} determines a partial orientation-preserving isometry of ρ(J) wich maps t′ into s′. Again, by Corollary 14.42, the elements w, w′ have the same axis L. This implies that γ(L) = L. This finishes the proof of (1). To prove (2) note that π fixes a point in L. Let π^{0} be the subgroup of index ≤ 2 in π which consists of elements preserving orientation on L. Then π^{0} acts trivially on L which concludes the proof. □

(iii) Output of the Process I: the thin case.

Suppose that some component C(Y) of Y has the thin type. Then we have an infinite sequence of band complexes X_{0}, X_{1}, ..., where for each i the complex X_{i+1} is obtained from X_{i} via the Process I applied to the minimal component C(Y). Let A_{n} denote the collection of subdivision annuli in X_{n} (see definition of the Move M5).

Proposition 14.71 Under the conditions above the group G splits over a subgroup that fixes an arc of T. The subgroups F_{i} are conjugate into vertex subgroups of this decomposition of G. Suppose that C(Y) has no zero weight bands, then the above decomposition is a free product.

Proof: It follows from the Proposition 14.59 (assertions 4 and 5) and Remark 14.52 that eventually there is a band B_{n} = b_{n} × I of weight 1 in C(Y_{n}) ⊂ X_{n} that meets the closure of X_{n} − (Y_{n} ∪ A_{n}) only in a subset of b_{n} ∪ dual(b_{n}). For any t > 0 there is a certain k ≥ 1 so that the intersection B_{n} ∩ Y_{n+k} consists of at least t components.

The band B_{n} is formed as the result of a collapse when the rule (R3) is applied.

Remark 14.72 Let D_{jn} be the subgraph in ∂Y_{n} which carries the subgroup F_{j} in H_{j} (in the relative case). The Remark 14.52 (part (2)) implies that D_{jn} ∩ B_{n} is contained in Γ(X_{n}).

Consider the horizontal segment h_{n} := b_{n}×{1/2} in B_{n}. It intersects certain subdivision annuli A_{1,n}, ..., A_{s,n} ⊂ A_{n} only in the end-points of h_{n}. Recall that A_{j,n} does not have to be an embedded annulus in X_{n}, what we actually have are continuous maps

α_{j,n} : 𝕊^{1} × [0, 1] → A_{j,n} ⊂ X_{n}

Let π : 𝕊^{1} × [0, 1] → [0, 1] denote the projection. For each j we have a finite set

P_{j,n} := α_{j,n}^{−1}(∂h_{n})

where distinct points have distinct projections to [0, 1]. Thus we get a disjoint collection of “horizontal” circles C_{j,n} := π^{−1}(π(P_{j,n})) in 𝕊^{1} × [0, 1]. We retain the notation C_{j,n} for the (homeomorphic) image of C_{j,n} in X_{n} and let

C_{n} := C_{1,n} ∪ ... ∪ C_{s,n} ⊂ X_{n}

Then Λ_{n} := h_{n} ∪ C_{n} is a finite graph (two bouquets of circles connected by a segment) which locally separates the complex X_{n}.

Then X_{n} is a graph of spaces with the edge space Λ_{n} and the vertex spaces the compactified components of X_{n} − Λ_{n}. This gives us a splitting of the fundamental group G of X_{n}. Every subgroup F_{j} in G is conjugate into one of the vertex subgroups of this decomposition of G, see Remark 14.72.

Our next goal is to show that this splitting is nontrivial.

First of all, generators of fundamental groups of the circles C_{j,n} are in the kernel of the action of H = π_{1}(C(Y)) on T_{H} (see Theorem 14.41). Thus the image of π_{1}(Λ_{n}) in G fixes T_{H} and hence fixes an arc of T.

If Λ_{n} does not separate X_{n} then G splits as an HNN-extension and we are done. Thus we assume that Λ_{n} separates into two components X′_{n} and X′′_{n}, in particular b ∩ dual(b) = ∅. Since B_{n} is eventually cut into arbitrary many bands and the number of long bands is bounded from above, there are paths p′ : I → X′_{n}, p′′ : I → X′′_{n} and long bands B̂′_{n}, B̂′′_{n} so that:

(a) p′(I), p′′(I) are contained in (nonsingular) leaves L′, L′′;

(b) L′ ⊂ B̂′_{n}, L′′ ⊂ B̂′′_{n};

Figure 52: Graphs corresponding to the subdivision annuli.

(c) Each intersection p′(I)∩B_{n} ⊂ b, p′′(I)∩B_{n} ⊂ dual(b) consists of two distinct points.

Therefore both paths p′, p′′ have nontrivial holonomy. Hence each vertex subgroup π_{1}(X′_{n}) and π_{1}(X′′_{n}) of the decomposition of G acts on T as a nontrivial group, therefore eacc vertex subgroup is nontrivial. On the other hand, the edge group (which is the image of π_{1}(Λ_{n}) in G) fixes T, hence it is a proper subgroup of G. This implies that the splitting of G is nontrivial and we are done. □

14.13 Proof of the decomposition theorem

In this section we assume that G is a finitely presentable group with a stable and minimal action on a tree T. Let F_{1}, ..., F_{n} be a collection of finitely generated subgroups of G each of which fixes a point in T.

Theorem 14.73 Under the above conditions the following is satisfied:

• If the action of G is not pure then G splits over a subgroup E so that E fits into a

short exact sequence

1 → K_{E} → E → C → 1

where K_{E} fixes an arc of T and C is either finite or cyclic. The subgroup E fixes a point in T. Each subgroup F_{j} is conjugate into one of the vertex subgroups of the decomposition of G.

• If the action is pure then G belongs to one of the following types:

(a) The surface type:

1 → K → G → π_{1}(Ø) → 1

where K is the kernel of the action of G on T and Ø is a 2-dimensional cone-type orbifold. Each of the subgroups F_{j} maps into a torsion or peripheral subgroup of π_{1}(Ø).

(b) The toral type: T is a line and

1 → K → G → A → 1

where K is the kernel of the action of G on T and A is a subgroup of Isom(ℝ). Each of the subgroups F_{j} maps into a torsion subgroup of A.

(c) The thin type: G splits over a subgroup that fixes an arc of T. Each subgroup F_{j} belongs to one of the vertex subgroups of the decomposition of G.

Proof: We start by constructing a band complex X as in Theorem 14.35 and then transform it to a band complex X′ which has which has no faults (see Lemma 14.44). The we apply the Rips Machine to each minimal component of the union of bands Y′ ⊂ X′.

If a component C(Y′) has thin type then we proceed as in Proposition 14.71: the image of π_{1}(C(Y)) in G admits the required splitting. This splitting is given by decomposition of the band complex X_{n} (obtained by applying Process I inductively to C(Y′)), as in Proposition 14.71, this decomposition yields the required splitting of the group G splits.

Hence in what follows we can assume that no component of Y′ has thin type. If a component C(Y′) is of surface or axial type then:

For each leaf L ⊂ C(Y) the image π of π_{1}(L) in G fits into a short exact sequence

1 → K_{L} → π → C → 1

where K_{L} fixes an arc in T and C a cyclic group (see Propositions 14.67, 14.70). Thus if X′ is pure then the assertion directly follows from Propositions 14.67, 14.70.

If X is not pure then G has a graph of groups decomposition (see Theorem 14.45 and Remark 14.46). Suppose that the decomposition is nontrivial. We already know that each edge group E of this decomposition has a fixed point on T. Each edge group E is contained in the image of the fundamental group of a leaf L of a component C(Y′).

If this component is minimal (i.e. has surface or axial type) then E satisfies the assertion of Theorem by Propositions 14.67, 14.70. Each simplicial component C(Y′) is an I-bundle over a leaf. By passing to a finite cover { \atop C(Y′)} over C(Y) we can assume that { \atop C(Y′)} is the trivial interval bundle over a leaf. Hence if D ⊂ C(Y′) is an edge-subset, then its fundamental group is a finite extension of a group which fixes an arc on T. □

Corollary 14.74 Suppose that the action of G is stable and arc stabilizers are virtually solvable. Then one of the following holds:

a If the action of G is not pure or is pure and thin then G splits over a subgroup E so that E is a finitely generated virtually solvable group which fixes a point in T. Each subgroup F_{j} is conjugate into one of the vertex subgroups of the decomposition of G.

b The surface case:

1 → a virtually solvable group → G → π_{1}(O) → 1

where O is a 2-dimensional cone-type orbifold. Each of the subgroups F_{j} maps into a torsion or peripheral subgroup of π_{1}(O).

c The toral case: G is virtually solvable.

14.14 Simplicial case of Skora’s theorem

Lemma 14.75 Let f : T → T′ be a morphism between simplicial G-trees where the tree T′ is small and the tree T satisfies the property:

If e, e′ are distinct edges incident to a common vertex v then the subgroup of G_{v} generated by stabilizers G_{e}, G_{e′} of e, e′ is not virtually nilpotent.

Then for every x, y ∈ T which belong to distinct edges we have: f(x) ≠ f(y).

Proof: Suppose that x, y ∈ T belong to distinct edges and f(x) = f(y). Connect x and y by a geodesic segment γ in T. The map f is a morphism, thus there exists a subdivision of γ into subsegments γ_{i} so that the restriction of f to each γ_{i} is injective. On the other hand, f|γ is not injective, therefore we can find two distinct edges e, e′ ⊂ T which share a common vertex v ∈ γ so that k = f(e ∩ γ) = f(e′ ∩ γ). It follows that the subgroups G_{e}, G_{e′} fix k which consists of more than one point. On the other hand, the subgroup of G generated by G_{e}, G_{e′} is not virtually nilpotent. This contradicts the assumption that the G-tree T′ is small. □

We now suppose that S is either a compact hyperbolic surface with geodesic boundary or a flat annulus, G = π_{1}(S), and T is a small simplicial G-tree which is elliptic relative to the collection of peripheral subgroups of G (i.e. every cyclic peripheral subgroup of G fixes a point in T). Let ψ : S̃ → T be a G-equivariant map which maps each boundary geodesic L to a fixed point t_{L} of the stabilizer G_{L} of L in G. Let T′ ⊂ T be the subtree which is the convex hull of ψ(∂S̃).

Proposition 14.76 There is an extension ϕ : S̃ → T′ of the map ψ and a finite measured lamination (L, μ) with blow up (L′, μ′) so that the fibers of ϕ are complementary regions of L̃′ and the leaves of L̃′. Here L̃′ is the lift of L′ to the universal cover S̃.

Remark 14.77 We will also use alternative reformulation this proposition in terms of meaured foliations: the fibers of ϕ are leaves of a G-invariant measured foliation F̃ on S̃ so that F := F̃/G is a rational measured foliation.

Proof: In each edge of the quotient graph Γ = T′/G we choose a midpoint, take the inverse image M of the set of these midpoints in the tree T. Pick a simplicial G-equivariant extension ψ : S̃ → T′ of the map ψ so that ψ is transversal to the subset M ⊂ T. Thus ψ^{−1}(M) is a simplicial properly embedded G-invariant submanifold in S̃ which is disjoint from ∂S̃. By construction, if G_{D} is the stabilizer of a complementary region D to ψ^{−1}(M), then G_{D} fixes a point in T. The quotient L := ψ^{−1}(M)/G is a union of simple loops in S. By cut-and-paste procedure we transform the map ψ to such that no loops in this collection are nil-homotopic and no pair of loops in L bounds an annulus, we retain the notation ψ for the resulting map, it is still transversal to M. Let L̃ denote the lift of L to S̃. Then the complementary regions to L̃ in S̃ have free nonebalian stabilizers each of which has a (unique) fixed point in T. Define the transverse measure μ to L as follows: if L is a loop in L whose inverse image L̃ ⊂ L̃ maps to the midpoint of edge e ⊂ T′ then we assign the transverse Dirac measure to L multiplied by the length of e. We now isotope (L, μ) to blown up measured geodesic lamination for which we retain the name (L, μ), we continue to assume that L is disjoint from ∂S.

Now let R be the dual tree to the lamination (L, μ). Construct G-equivariant map f : R → T by mapping each vertex v (corresponding to a complementary region D_{v}) to the unique fixed point t_{v} in T of the stabilizer G_{D_{v}}. Map edge [vw] of R to the geodesic segment in T connecting t_{v} to t_{w}; it is clear that d(t_{v}, t_{w}) = d(v, w) so we choose the map [vw] → [t_{v}t_{w}] to be an isometry. Thus by Lemma 14.75 the morphism f is injective.

□

In the above proposition the isometry R → T′ is note necessarily an isomorphism of trees since we can arbitrarily subdivide edges of the quotient graph Γ and thus add extra vertices to the tree T′. However we simply pull-back these extra vertices via the isometry f to the tree R and get a new simplicial tree R′.

The tree R′ corresponds to a collection of simple disjoint loops none of which is nilhomotopic. However some of the new loops are mutually isotopic and some could be boundary parallel.

Corollary 14.78 There is a system of disjoint simple homotopically nontrivial loops α_{1}, ..., α_{n} so that the simplicial G-tree dual to the decomposition of the universal cover of S by inverse images of the loops α_{i} is isomorphic to the simplicial G-tree T.

14.15 Proof of Skora’s theorem in the case of geometric group action

Definition 14.79 Let T be a G-tree where G is a finitely-presentable group. The action of G on T is called geometric if it admits an exact resolution via the universal cover of a band complex. Otherwise the action is called nongeometric.

We first consider this notion in the case of simplicial trees T. Then clearly action of G on T is geometric iff all vertex and edge stabilizers are finitely-generated groups. This immediately gives us an example of a nongeometric action. Namely, let S be a closed hyperbolic surface ϕ : π_{1}(S) = G → ℤ is a nontrivial homomorphism. Clearly kernel N of ϕ is infinitely generated. Then G is an HNN-extension of N which gives us an action of G on the simplicial tree ℝ (vertices are the integers), the action of G on ℝ factors through ℤ acting on ℝ by integer translations. Vertex and edge stabilizer subgroups of this action are infinitely generated (they are all equal to N). Hence this is a nongeometric action.

There are more complicated examples of nongeometric free actions of groups on trees (when G is a finitely generated free group), we refer to [70] for details.

In this section we prove Skora’s theorem under the extra assumption that the group action is geometric:

Theorem 14.80 Let S be a compact hyperbolic surface with the fundamental group G and the boundary components α_{1}, ..., α_{m}, F_{i} := ι_{i∗}(π_{1}(α_{i})), i = 1, ..., m, where ι_{i} : α_{i} ↪ S is the inclusion. Consider a small minimal geometric action of G on a tree tree T such that each subgroup F_{i} fixes a point in T (i = 1, ..., m). Then there is a measured foliation G on S such that the dual G-tree T_{G} is isomorphic to the G-tree T.

Proof: Let X be a band complex whose universal cover gives an exact resolution to the Gtree. Apply Preliminary Motions of the Rips Machine to the band complex X (see section 14.10), we retain the name X for the resulting band complex. Let Y ⊂ X be the union of bands in X. We apply the results of Sections 14.12, 14.13 to X.

Let C(Y) be a minimal component of Y and B ⊂ C(Y) a weight zero band. Then the fundamental group π_{1}(B) of the annulus corresponding to B acts trivially on the tree T_{H} where H is the image of the fundamental group of C(Y) in G (see Theorem 14.41). Thus, since the action of G on T is small, it follows that the image of π_{1}(B) in G is trivial. Let b be a base of such band b and r : X → cl(X −B) the retraction whose restriction to the band B is the map (x, y) ↦ x. Therefore the map r induces an isomorphism of fundamental groups and we can eliminate all the weight zero bands from minimal components of Y (Move 0) without changing the dual tree. We retain the name X for the resulting complex.

Lemma 14.81 Each minimal component of Y has the surface type.

Proof: (1) Suppose a minimal component C(Y) has thin type. Then by Proposition 14.71 the group G splits as a nontrivial free product and this splitting is relative to the collection of the subgroups F_{i}, i = 1, ..., m. This contradicts the assumption that F_{1}, ..., F_{m} represent the fundamental groups of all boundary circles of S.

(2) Suppose a minimal component C(Y) has the toral type. According to Lemma 14.69, Proposition 14.70 the group H, which is the image of π_{1}(C(Y)) in G, has an invariant geodesic line L ⊂ T and this action has nondiscrete Bass group. Therefore H contains an abelian subgroup of rank 2, which contradicts the assumption that G is the fundamental group of a hyperbolic surface. □

Let C(Y) be a minimal component of Y. We know that C(Y) is homeomorphic to a compact surface. The measured foliation on Y restricts to a measured foliation on this subsurface. Our next goal is to understand how 2-cells Δ^{2} of X are attached to the boundary loops of C(Y). Without loss of generality we can assume that every attaching map ∂Δ^{2} → ∂C(Y) has nonzero degree.

Suppose that λ is a boundary loop of C(Y) which is nil-homotopic in X. Then λ necessarily splits X into several connected components (otherwise G would split into a free product relative to the collection F_{1}, ..., F_{m}). Let Z be a component of cl(X −λ) which does not contain C(Y). Therefore the image of π_{1}(Z) in G is trivial, thus Z does not compain any minimal components of Y and we replace Z by a disk attached along λ without changing the fundamental group of X and the dual tree. We do it with respect to each loop λ as above, we will retain the notation X for the resulting complex.

Lemma 14.82 Let λ be a boundary loop of C(Y) such that there is a 2-cell in X attached along λ. Then the attaching map has degree ±1.

Proof: For each boundary loop λ of C(Y) which has a 2-cell attached to it we pick a generator γ of π_{1}(λ) and let m_{λ} denote the degree of the attaching map. We retain the name γ for the image of γ in π_{1}(Y). Consider the kernel K of the map π_{1}(C(Y)) → π_{1}(X) = G. The subgroup K is normally generated by the elements γ^{m_{λ}}. If at least one m(λ) is different from ±1 then the image H of π_{1}(C(Y)) in G is the fundamental group of a 2-dimensional orbifold of cone type with nonempty singular locus. Thus either H has torsion or it is a trivial group. Neither case is possible since G is torsion-free and minimality of C(Y) implies that π_{1}(C(Y)) does not act on T as the trivial group. □

Lemma 14.83 Let λ be a boundary loop of C(Y) such that there is a 2-cell in X attached along λ. Then λ contaions at least two singular points of the measured foliation on C(Y).

Proof: Since λ is a boundary loop and the component C(Y) is minimal, it follows that λ contains at least one singular point x. The point x is necessarily contained in the interior of a base b of a band B in Y. If x is the only singular point on λ then there is a cancatenation of bands B_{1} = b_{1} × I, B_{2} = b_{2} × I, ..., B_{s} = b_{s} × I so that:

• The bottom of B_{1} is a base b_{1} ⊂ Γ which has x as an end-point; the top of B_{s} is a base dual(b_{s}) ⊂ Γ which has x as an end-point;

• (b_{1} ∪ dual(b_{s})) ∩ b forms a neighborhood of x in b;

• λ − ∪_{i=1}^{s}b_{i} is the disjoint union of open segments (x_{i}, x_{i+1}) ⊂ ∂B_{i}, x_{1} = x.

Thus we can find a vertical path α in B_{1} ∪ ... ∪ B_{s} near λ which starts at b_{1} and ends at dual(b_{s}). Since the loop λ is nil-homotopic in X, the path α lifts to a path α̃ in the universal cover X̃ so that the end-points of α̃ are contained in a lift b̃ of the base b. This implies that the resolution ρ : X̃ → T is not injective on b̃ which contradicts the assumption (A4) (Section 14.10). □

Let C_{i}(Y) be a minimal component of T, define C_{i}(X) by adjoining to C_{i}(Y) all 2-cells from X which are attached along the boundary circles of C_{i}(Y). Note that π_{1}(C_{i}(X)) injects into G. Let Σ_{i} be the surface obtained from C_{i}(X) by collapsing each 2-cell (that is not in Y) to a point. The singular measured foliation on C_{i}(Y) projects to a singular measured foliation on the surface Σ_{i} (since every boundary circle of C_{i}(Y) which is nil-homotopic in X contains at least 2 singular points).

Remark 14.84 This would conclude the proof of Theorem 14.80 in the case when the action of G on T is pure. The main objective of the rest of the argument is to combine “pure” and “simplicial” components of X into a measured foliation of a surface so that “pure” components correspond to spiral regions and “simplicial” components correspond to rational regions.

We replace each C_{i}(X) in X by the corresponding surface Σ_{i}. Our next goal is to make analogous replacement for components of cl(X − ∪_{i}Σ_{i}). The decomposition of X into the union of components of ∪_{i}Σ_{i} and components of cl(X − ∪_{i}Σ_{i}) corresponds to graph of groups decomposition of the group G where edge subgroups are infinite cyclic. Therefore by Corollary 14.78 we can realize this decomposition of G by geometric decomposition of the surface S along a collection of disjoint closed curves, none of which is nil-homotopic (however some of these curves could be homotopic to each other). We let S_{i} denote the components of this decomposition of S. We will colour the components S_{i} in two colours: black if they correspond to Σ_{i} ⊂ X and white if they do not. Clearly no two components of the same colour are adjacent to each other.

For each surface Σ_{i} ⊂ X the peripheral subgroups of π_{1}(Σ_{i}) correspond to the edge subgroups incident to the group π_{1}(Σ_{i}) in the graph of groups decomposition of G = π_{1}(X). Let S_{i} denote the subsurface of S whose fundamental group corresponds to π_{1}(Σ_{i}) under the isomorphism π_{1}(S) → π_{1}(X). Then the isomorphism π_{1}(S_{i}) → π_{1}(Σ_{i}) preserves the peripheral structure (since it is given by the incident edge subgroups), therefore it is induced by a homeomorphism S_{i} → Σ_{i}. We pull-back the measured foliation from Σ_{i} to S_{i}. Let S̃_{i} be a lift of S_{i} to the universal cover S̃ of S. The quotient S̃_{i}/ ≅ by the lift of the measured foliation from S_{i}, is a metric tree acted upon by π_{1}(S_{i}); this tree has canonical injective equivariant map into the tree T which is given by restriction of the reslution ρ : X̃ → T to the appropriate lift of Σ_{i}.

We now consider a white component S_{j} in the decomposition of S. Let S̃_{j} be a lift of S_{j} to S̃. Then we have an equivariant map

ψ : ∂S̃_{j} → T

which is given by restriction of the maps of adjacent black components of S̃ to their boundary arcs. Using Proposition 14.76 we construct an equivariant extension ϕ of this map to S̃_{j} so that the fibers are leaves of a measured foliations on S_{j}. The image of the extension ϕ is contained in the convex hull of ϕ(∂S̃_{j}). This implies that ϕ(S̃_{j}) ⊂ T_{Z_{j}} where T_{Z_{j}} is the dual tree to the foliation of the component Z_{j} ⊂ X corresponding to S_{j}.

We let F be the generalized measured foliation on the surface S which is the union of measured foliations of black and white components. The foliation F has the properties:

• There is a map ϕ : S̃ → T which induces a representation p_{∗} : π_{1}(S) → Isom(T) so that p_{∗} = ι ∘ q_{∗} where ι is the isomorphism π_{1}(S) → π_{1}(X).

• The fibers of ϕ are the leaves of the measured foliation F̃, which is the lift of F to X̃.

This concludes the proof of Theorem 14.80. □

14.16 Geometric and nongeometric actions on trees

The main objective of this section is to prove Skora’s theorem in general case. Recall that if the tree T is simplical then according to Proposition 14.76 the tree T is dual to a rational measured rational foliation on S.

Definition 14.85 Suppose that we have a collection of G-trees T, T_{n}, n = 1, 2, 3, .... Then G-trees T_{n} are said to converge strongly to the G-tree T if the following is satisfied:

• There is a sequence of G-morphisms f_{n} : T_{n} → T, f_{n,k} : T_{n} → T_{k}, k ≥ n;

• f_{n} = f_{k} ∘ f_{n,k};

• For each n and each pair of points x, y ∈ T_{n} there is N = N(n, x, y) so that for all

k ≥ N we have:

d(f_{n,k}(x), f_{n,k}(y)) = d(f_{n}(x), f_{n}(y)) .

Lemma 14.86 If the G-tree T is small then for sufficiently large n the G-tree T_{n}.

Proof: The proof is obvious and is left to the reader. □

Theorem 14.87 (Levitt, Paulin, [124]) Suppose that we have a finitely-presentable group G and a G-tree T (relative to a collection of subgroups F_{i}). Then T is a strong limit of minimal geometric actions of G on trees T_{n} (relative to the collection of subgroups F_{i}). If T is small then T_{n} are also small for all sufficiently large n. If T is minimal then T_{n} are also minimal for all sufficiently large n. Moreover, if X_{n} is the band complex which gives an exact resolution of the G-tree T_{n} then there exists a continuous foliation-preserving map ϕ_{n,n+i} : X_{n} → X_{n+i} which induces an isomorphism of the fundamental groups.

Proof: We apply Theorem 14.35 to construct a band complex X and (relative with respect to the collection of subgroups F_{j}) resolution ρ : X̃ → T of the action of G on T, where X̃ is the universal cover of X. Recall that in the construction of X we are free to choose a finite compact subtree K ⊂ T containing a certain finite set. We choose an increasing sequence K_{n} of such subtrees which exhaust the tree T. According to the construction in Theorem 14.35 the tree K_{n} is the graph on which the band complex X_{n} is based. Moreover, if K̃_{n} is a component of the lift of K_{n} to X̃_{n}, then the resolution ρ_{n} : X̃_{n} → T embeds K̃_{n} isometrically into T.

By construction each group F_{i} fixes a point in T_{n}. Now we need to verify that minimality of T implies minimality of T_{n}. By construction the G-orbit of the subset K̃_{1} ⊂ T_{n} is the whole tree T_{n}. Thus it suffices to show that (for sufficiently large n) each point of K̃_{1} lies in an axis of an axial element of G acting on T_{n}. Since the G-tree T is minimal it follows that each point of K_{1} ⊂ T is contained in the axis A_{g} of an axial isometry g. Let D_{g} ⊂ A_{g} be the union of a fundamental domain F_{g} for the action of ⟨g⟩ on A_{g} and g(F_{g}). We choose K_{n} such that

D_{g_{i}} ⊂ K_{n}, i = 1, ..., ℓ, where K_{1} ⊂ ∪_{i=1}^{ℓ}D_{g_{i}}

Now lemma 11.7 implies that each element g_{i}, i = 1, ..., ℓ acts as an axial isometry on T_{n} and the axis of g_{i} contains the image of K̃_{1} in T_{n}. Therefore the G-tree T_{n} is minimal.

The isometric embeddings K̃_{n} ↪ T_{n+i}, K̃_{n} ↪ T determine a collection of morphisms

f_{n,n+i} : T_{n} → T_{n+i}, f_{n} : T_{n} → T

as follows. The inclusion K_{n} ↪ K_{n+i} extends to continuous foliation-preserving map ϕ_{n,n+i} : X_{n} → X_{n+i} which induces an isomorphism of the fundamental groups. The map f_{n,n+i} is the projection of the lift of ϕ_{n,n+i} to the quotient trees. (According to the construction in Theorem 14.35 each band in X_{n} is also a band in X_{n+i}.) We conclude that the action of π_{1}(S) on T is the strong limit of geometric actions as required by Theorem 14.87.

□

Proposition 14.88 The morphisms f_{n,k} are isomorphisms for all sufficiently large n.

Proof: Firstly, strong convergence implies that for each element g ∈ G there is N so that for all n ≥ N, k ≥ n we have equality of the translational lengths:

ℓ_{T_{n}}(g) = ℓ_{T_{k}}(g) = ℓ_{T} (g)

By Corollary 12.27 there is a finite collection g_{1}, ...g_{m} of elements of G so that the equivalence class of each measured foliation F is uniquely determined by the translational lengths

(ℓ_{T_{F}} (g_{1}), ..., ℓ_{T_{F}} (g_{m}))

It follows that all the G-trees T_{n} are isomorphic^{12} for sufficiently large n and they are determined by the same measured geodesic lamination L on the hyperbolic surface S. Now suppose that

f := f_{n,k} : T_{n} = T_{L} → T_{k} = T_{L}

is a nontrivial folding, i.e. there is an arc [x, y] ⊂ T_{n} so that f(x) = f(y). Let L̃ be a measured geodesic lamination in ℍ^{2} which is a lift of L to the universal cover of S. We lift [x, y] to a hyperbolic geodesic segment [x̃, ỹ] in ℍ^{2} so that the the projection

[x̃, ỹ] → [x, y]

is an isomorphism with respect to the transversal measure induced by L (on [x̃, ỹ]) and the Lebesgue measure induced on [x, y] by the metric on T. By making appropriate choice of x, y we may assume that [x̃, ỹ] do not belong to L̃. Let L̃_{1} be the sublamination in L̃ which consists of geodesics which intersect [x̃, ỹ] nontrivially. The infinite hyperbolic geodesic which contains [x̃, ỹ] can be approximated by axes of hyperbolic elements of G, hence there exists a hyperbolic element g ∈ G with the axis A so that the transversal measure of the intersection A ∩ L̃_{1} is the same as the transversal measure of the intersection [x̃, ỹ] ∩ L̃. Recall that f(x) = f(y) ∈ T_{k}, hence the translational length of g in T_{k} is strictly smaller than the translational length of g in T_{n}. This is a contradiction.

Now we can finish the proof of Skora’s theorem in the general case. Using the Proposition 14.88 we conclude that for each pair of points x, y ∈ T_{1} we have:

d(x, y) = d(f_{n,k}(x), f_{n,k}(y)) = d(f_{n}(x), f_{n}(y))

for sufficiently large n, k. This implies that f_{n} : T_{n} → T is an isomorphism for all sufficiently large n. Therefore T is dual to a measured foliation on the surface S. □

12Apriori such isomorphism is not given by the maps f_{n,k}.

14.17 Application of Rips’ theory to compactness properties of representation varieties

Proof of Theorem 14.1.

Suppose that (M, P) is a compact orientable paired Haken 3-manifold, which has incompressible boundary surface ∂_{0}M and the group π_{1}(M) = Γ is not almost Abelian. We will call such paired manifold semi-simple. Note that each almost solvable subgroup of Γ is either trivial, or ℤ, or a peripheral subgroup ℤ^{2}. Consider an effective, isometric, minimal, small, nontrivial action of Γ on a tree T so that the fundamental group H_{j} of each component of P has a fixed point on T. Such action is necessarily stable by Proposition 14.10. Thus we apply Corollary 14.74.

Step 1. Consider the cases (b) and (c) of a surface and axial actions Γ ↷ T. Then the group Γ either a surface group and for each component P_{i} ⊂ P, π_{1}(P_{i}) is a peripheral subgroup of Γ or Γ is abelian. In both cases the manifold M is homeomorphic to an interval bundle over a compact surface Σ so that the designated parabolic locus P is the I-bundle over ∂Σ. Thus M itself is the window and Theorem 14.1 trivially follows.

Step 2. Now we consider the generic case (a): nontrivial decomposition of π_{1}(M). Since M is a paired manifold, each edge subgroup C for this decomposition is either cyclic or trivial. The case of trivial subgroup means that ∂M − P is compressible (Theorem 2.35) which contradicts our assumptions. Thus C is cyclic and according to Theorem 2.36 there is an essential annulus A ⊂ M, which is disjoint from P. Then A (up to isotopy) is contained in the window of M and we split M open along A. The resulting manifold M′ consists either of one or two components, its boundary contains two annuli A_{1}, A_{2} corresponding to A. Note that each π_{1}(A_{j}) has a fixed point on T. We let P′ := P ∪ A_{1} ∪ A_{2} be the new designated parabolic locus for M′. It is easy to see that (M′, P′) is again a paired manifold, so we repeat this procedure again and again. The process of decomposition of (M, P) along essential annuli will terminate after finitely many steps when we get a manifold M_{k} (the Euler characteristic to ∂M is increasing on each step). Note that on each step we split only window components. Thus, each component of (M, P) − W is a component of the complement to the window V of M_{k}. Let Q ⊂ ∂M_{k} be the designated parabolic locus that we have got in the process (it includes P and the annuli that we have created after splitting our manifolds).

Step 3. Let N be a component of M_{k}. We need to prove that the fundamental group of each component of N − V has a fixed point on T. If π_{1}(N) has a global fixed point on T then we are done. So, suppose that we have a nontrivial action of π_{1}(N) on a minimal subtree T′ ⊂ T which is π_{1}(N)-invariant. The only case that we can have: this action is of the “surface type”. However we have already considered this case in Step 1, when we concluded that N itself is the window in (N, Q ∩ N).

This concludes the proof of Theorem 14.1. □

Corollary 14.89 Suppose that (M, P) is a simple paired acylindrical 3-manifold with the fundamental group G, T is a small G-tree so that the action G ↷ T is parabolic (with respect to P). Then G has a global fixed point on T.

Proof: Directly follows from Theorem 14.1 since W = ∅. □

Corollary 14.90 Suppose that (M, P) is a simple paired 3-manifold with the fundamental group G, Γ is a group which contains G as a subgroup of finite index, Γ ↷ T is a small action of Γ on a metric tree T so that the restriction of this action to G is parabolic. Then Γ has a global fixed point on T.

Proof: This directly follows from Corollary 14.89 and the following general

Proposition 14.91 Suppose that G ↷ X is an isometric action of a group G on a CAT (0)- space which has a global fixed point x. Then for every extension of this action to a group Γ which contains G as a subgroup of finite index, Γ also has a global fixed point on X.

Proof: Without loss of generality we can assume that G is normal in Γ. Then the Γ-orbit of the point x is a finite set {x_{1}, ..., x_{n}}. Take x_{0} be the barycenter of this set as in Section 4.2. Then x_{0} is Γ-invariant. □

Theorem 14.92 (Relative compactness theorem, W. Thurston, J. Morgan, P. Shalen.) Suppose that (M, P) is a simple acylindrical paired 3-manifold. Then the relative representation variety

D(π_{1}(M), π_{1}(P); P SL(2, ℂ))

is compact.

Proof: Suppose not, then there is a minimal nontrivial parabolic action of G on a tree T. However Corollary 14.89 implies that G must have a global fixed point on T. Contradiction. □

Similarly we get:

Theorem 14.93 Suppose that (M, P) is a simple acylindrical paired 3-manifold, Γ is a group which contains G = π_{1}(M) as a subgroup of finite index. Then the relative representation variety D(Γ, π_{1}(P); P SL(2, ℂ)) is compact.

15 Outline of the Proof of the Hyperbolization Theorem

Finally I can present an outline of the proof of Thurston’s Hyperbolization Theorem. By Theorem 9.22 it is enough to prove the Hyperbolization Theorem for a finite covering of M, thus we will consider only compact orientable paired manifolds (M, P), which contain orientable superincompressible surfaces and where P = ∂M is the designated parabolic locus. Thurston’s theorem is proven by induction on levels of the Haken hierarchy of M. It turns out that the main problem in proving Theorem 2.44 is the last step of induction:

Theorem 15.1 (The last step of induction.) Suppose that (M, P) is an orientable paired atoroidal manifold, whose boundary P consists of incompressible tori. Suppose that Σ_{M} ⊂ M is an orientable superincompressible surface which consists of either one or two components, so that the paired manifold (N, Q) obtained by splitting M along Σ_{M}, consists either of one or two components which admit geometrically finite hyperbolic structures with the parabolic locus Q. In the case when Σ_{M} consists of two components we assume that none of the components of M − Σ_{M} is an interval bundle over a surface. Then the paired manifold (M, P) admits a complete hyperbolic structure of finite volume.

There is a difference between this theorem and one formulated in [150], which deals with connected superincompressible surfaces only. The reason for necessity to consider disconnected superincompressible surfaces will become clear in Section 19.6.

Case A. The surface Σ_{M} is not a virtual fiber in a fibration of M over 𝕊^{1} (our assumptions imply that this is always the case provided that Σ_{M} is disconnected). It may happen that still one of the components of (M − Σ_{M}, P − Σ_{M}) is an interval bundle over a surface. In this case we just take a double covering over M so that a component of the lift of Σ_{M} does not separate (see Section 2.9). Thus we can assume that in the Case (A) none of the components of (M − Σ_{M}, P − Σ_{M}) is an interval bundle over a surface.

Case B. M (or its double cover) is fibered over a circle and Σ_{M} is a (virtual) fiber. This is exactly the case when N is an interval bundle over the a surface which is either homeomorphic to Σ_{M} or is 2-fold covered by Σ_{M}.

Steps of the proof:

Step 0. “The orbifold trick”. We shall postpone the discussion of this step for a while (Chapter 19), since it requires introduction of new objects: orbifolds. The main theorem to be proven in the Chapter 19, Sections 19.4–19.6, is the following:

Theorem 15.2 Assume that Theorem 15.1 (Case A) is valid for all (M, P). Suppose that (M′, P′) is a compact paired manifold with the designated parabolic locus P′, χ(M′) ≠ 0; L′ ⊂ M′ is an orientable connected superincompressible surface. Let (M, P) be the manifold obtained by splitting (M′, P′) along L′, the surface P ⊂ ∂M consists of a collection of simplyconnected components as well as a union T of some tori and annuli. Suppose that the paired manifold (M, T) admits a geometrically finite hyperbolic structure with the parabolic locus T. Then the paired manifold (M′, P′) also admits such structure (where P′ is the parabolic locus).

Theorems 15.2 and 15.1 (Case A) imply the following:

Theorem 15.3 Let (M, P) be an orientable paired atoroidal manifold, so that χ(∂M = P) = 0. Suppose that Σ_{M} ⊂ M is an orientable connected superincompressible surface, so that the paired manifold (N, Q) obtained by splitting M along Σ_{M} is not an interval bundle over a surface (equivalently, Σ_{M} is not a virtual fiber in a fibration over 𝕊^{1}). Then the paired manifold (M, P) admits a complete hyperbolic structure of finite volume.

Proof: Consider a Haken hierarchy for (M, P): (M^{1}, P^{1}) := (N, Q), (M^{2}, P^{2}) ,..., M^{h}. Here M^{h} is a collection of balls; M^{j+1} is obtained from (M^{j}, P^{j}) by splitting along a connected superincompressible surface Σ^{j} and P^{j+1} is obtained from P^{j} − Σ^{j} by removing contractible components. Clearly M^{h} admits a convex hyperbolic structure (take a collection of round balls in ℍ^{3}). Going backwards along the hierarchy and using Theorem 15.2 we construct geometrically finite hyperbolic structures on each (M^{j}, P^{j}), including (M, P). □

Note that in the process of the proof we also derive Theorem 2.45 from Theorems 15.1 and 15.2. In Section 19.7 we will derive Theorem 2.45 directly from Thurston’s Hyperbolization Theorem.

Proof in the case (B) is quite different and we first concentrate on the generic Case A, leaving discussion of the exceptional Case (B) to the Section 15.2.

15.1 Case A: The generic case

Proof of Theorem 15.1 (Case A). It is very inconvenient to keep the distinction between the cases when Σ_{M} is connected and disconnected. Instead we formulate a theorem which covers both connected and disconnected cases.

Theorem 15.4 Suppose that (N, Q) is a paired Haken 3-manifold (which consists either of one or of two components), we assume that ∂_{0}N is incompressible, (N, Q) admits a geometrically finite hyperbolic structure and no component of (N, Q) is an interval bundle over a surface. Suppose that Σ_{1}, Σ_{2} are components of ∂_{0}N and τ : Σ_{1} ∪ Σ_{2} → Σ_{2} ∪ Σ_{1} is a gluing involution such that the manifold M := N/τ is atoroidal, τ(Σ_{1}) = Σ_{2}. Let P denote the image of Q in N/τ. Then (N/τ, P) admits a geometrically finite hyperbolic structure.

Note that once Theorem 15.4 is proven, Theorem 15.1 (Case A) follows by induction (which has either length 1 or 2). Thus, we are proving Theorem 15.4.

I recall that for a Kleinian group G ⊂ P SL(2, ℂ) we use the notation \overline{M}(G) for the manifold with boundary (ℍ^{3}∪Ω(G))/G. The compact manifold \overline{M}(G)^{0} is the complement to the union of open Margulis cusps. Recall that ∂_{P} \overline{M}(G)^{0} is the intersection of the boundary of \overline{M}(G)^{0} with the union of Margulis cusps in M(G). D(G, P SL(2, ℂ)) is the space of conjugacy classes of discrete and faithful representations of G into P SL(2, ℂ). The relative representation space

D(G, π_{1}(Q); P SL(2, ℂ)) ⊂ D(G, P SL(2, ℂ))

consists of (conjugacy classes) of representations whose restrictions to π_{1}(Q_{j}) are parabolic for each components Q_{j} of Q.

According to the hypothesis, there exists a geometrically finite Kleinian group G (or a pair of such groups) such that (N, Q) = (\overline{M}(G)^{0}, ∂_{P} \overline{M}(G)^{0}) (in the case when N is connected), otherwise N = N_{1} ∪ N_{2} = \overline{M}(G_{1})^{0} ∪ \overline{M}(G_{2})^{0} and G = (G_{1}, G_{2}). In any case the boundary of \overline{M}(G) contains Σ_{1} ∪ Σ_{2} = Σ. Let τ_{1} : Σ_{1} → Σ_{2}, τ_{2} : Σ_{2} → Σ_{1} be the pair of gluing maps so that τ_{2} = τ_{1}^{−1}, τ = τ_{1} ∪ τ_{2}. The Teichmüller space of G is the product of Teichmüller spaces of the boundary components ∂_{0}N and it is too large for us. So, we shall fix once and for all marked conformal structures on all components of ∂_{0}N except Σ, the resulting subvariety in T (G) will be denoted T_{Σ}(G) (if N is connected); in the case when N consists of two components we get T_{Σ_{j}}(G_{j}) ⊂ T (G_{j}) and T_{Σ}(G) := T_{Σ_{1}}(G_{1}) × T_{Σ_{2}}(G_{2}). Clearly T_{Σ}(G) ≅ T (Σ).

Step 1. (N, Q) is a semisimple paired manifold, thus the surfaces Σ_{j} correspond to quasifuchsian subgroups F_{j} ⊂ G, let Ω_{j} denote the component of Ω(F_{j}) which belongs to Ω(G), j = 1, 2 (or to Ω(G_{j}) if N is not connected). We use the notation ϕ for the isomorphism of the groups F_{1} → F_{2} induced by τ_{1}.

Consider the case when N is connected.

Theorem 15.5 In order to prove Theorem 15.4, it is enough to find a point [ρ] ∈ T_{Σ}(G) induced by a quasiconformal map f such that: for Ω′_{j} = f(Ω_{j}) and Ω̄′_{j} = \overline{ℂ} − Ω′_{j}, there is a Moebius transformation A ∈ P SL(2, ℂ)

A : Ω′_{1} → Ω̄′_{2}, A : Ω̄′_{1} → Ω′_{2}

which induces the isomorphism ρϕρ^{−1} between F′_{1} = ρ(F_{1}) and F′_{2} = ρ(F_{2}).

Remark 15.6 The group uniformizing the manifold N/τ is the HNN extension of ρ(G) via the Moebius transformation A.

In the case when N is not connected Theorem 15.5 should be reformulated as follows.

Theorem 15.7 In order to prove Theorem 15.4, it is enough to find a pair of points [ρ_{j}] ∈ T_{Σ_{j}}(G_{j}) induced by quasiconformal maps f_{j} such that: for Ω′_{j} = f_{j}(Ω_{j}) and Ω̄′_{j} = \overline{ℂ} − Ω′_{j} there is a Moebius transformation

A : Ω′_{1} → Ω̄′_{2}, A : Ω̄′_{1} → Ω′_{2}

which induces the isomorphism ρ_{2}ϕρ_{1}^{−1} between F′_{1} = ρ_{1}(F_{1}) and F′_{2} = ρ_{2}(F_{2}).

Remark 15.8 In this case the amalgamated free product of ρ_{1}(G_{1}) and A^{−1}G′_{2}A is the group uniformizing the manifold M.

Step 2. On this step, given the gluing map τ : Σ → Σ, we construct a map σ = σ_{τ} : T (Σ) → T (Σ) such that existence of ρ in Theorems 15.5, 15.7 can be reformulated as follows.

Theorem 15.9 (Fixed Point Theorem) The map σ has a fixed point X ∈ T (Σ).

Remark 15.10 If X is a fixed point, then the image of X under the Bers bijection β : T (Σ) → T_{Σ}(G) is the representation ρ required by Theorems 15.5 and 15.7.

In order to prove Theorem 15.9 we establish two facts about the map σ.

Step 3.

Theorem 15.11 The map σ is a contraction of the Teichmüller metric.

Step 4. Via Theorem 15.11 we prove here that the Fixed Point Theorem follows from the the following “Weak Bounded Image Theorem”.

Theorem 15.12 (Weak Bounded Image Theorem.) The sequence σ^{n}(Y) is precompact in T (Σ) for every point Y ∈ T (Σ).

There is a stronger version of this result in the case when the paired manifold (N, Q) is acylindrical. We define \overline{Σ} to be the quotient Ω̄(F_{1})/F_{1} ∪ Ω̄(F_{2})/F_{2}.

Theorem 15.13 (Strong Bounded Image Theorem.) Consider the embedding β : T (Σ) → T_{Σ}(G) ↪ T (F_{1}) × T (F_{2}) = T (Σ) × T (\overline{Σ}). Then the projection of β(T (Σ)) to T (\overline{Σ}) is precompact.

Detailed discussion of the Steps 1 through 4 occupies the Chapter 16. Proof of the Weak Bounded Image Theorem is “the heart of the Hyperbolization Theorem”, this proof breaks into two major steps, which we discuss in details in the Chapter 17.

Step 5.

Theorem 15.14 For every Y ∈ T (Σ) the sequence β(σ^{n}(Y)) is precompact in the representation space

D(G, π_{1}(Q); P SL(2, ℂ)) ⊂ D(G, P SL(2, ℂ))

Remark 15.15 In the case when (N, Q) is acylindrical, this step is simplified by the fact that the relative representation space D(G, π_{1}(Q); P SL(2, ℂ)) is compact.

Step 6. Given Theorem 15.14, we have to prove that the sequence β(σ^{n}(Y)) is precompact not only in D(G, P SL(2, ℂ)) but also in T_{Σ}(G). Suppose that

ρ ∈ lim_{n→∞}[β(σ^{n}Y) = ρ_{n}]

What we have to prove is:

(1) The restriction of ρ to each F_{j} ⊂ π_{1}(Σ) has no accidental parabolic elements;

(2) The restriction of ρ to each F_{j} is geometrically finite.

It follows that the images ρ(F_{j}) are quasifuchsian (see Theorem 9.15). Then we apply Theorem 9.31 to conclude that for large n the representations ρ_{n} belong to T (ρG); therefore ρ ∈ T (G). Whence, by Corollary 9.32, the sequence ρ_{n} is convergent to ρ in the space T (G).

This finishes the proof of the Hyperbolization Theorem in the generic case.

15.2 Case B: Manifolds fibered over the circle

It follows from Theorem 9.22 and our results in Section 2.9 that it is enough to consider the Case (B) when the surface Σ_{M} itself is a fiber in a fibration of M over 𝕊^{1}.

Outline of Thurston-Otal approach:

Suppose that M is a compact 3-manifold fibered over 𝕊^{1}, Σ_{M} is a superincompressible surface as above, S := int(Σ_{M}). Let F denote the Fuchsian fundamental group of N, where N = M − Nbd(S). We use the notation τ for the gluing map τ : ∂N → ∂N. This map corresponds to an “aperiodic” homeomorphism τ : S → S, which means that for any homotopically nontrivial nonperipheral loop γ ⊂ S and any n ≠ 0 the loops τ^{n}(γ) and γ are not freely homotopic (see Section 2.9). The surface S admits a complete hyperbolic metric of finite area which is uniformized by a Fuchsian group F ⊂ P SL(2, ℝ).

Step 7.

Theorem 15.16 (Compactness theorem for aperiodic homeomorphisms.) For any aperiodic homeomorphism τ : S → S and α ∈ T (S), the sequence

β([ρ_{n}]) := (τ^{n}(α), τ^{−n}(α)) ∈ T (S) × T (\overline{S}) = T (F)

is subconvergent in D(F, P SL(2, ℂ)).

Let ρ_{∞} is a limit of the sequence of representations ρ_{n}, Γ_{∞} := ρ_{∞}(F).

Theorem 15.17 The limiting group Γ_{∞} has empty discontinuity domain and there is a quasiconformal homeomorphism f_{∞} : \overline{ℂ} → \overline{ℂ} so that:

• f_{∞}Γ_{∞}f_{∞}^{−1} = Γ_{∞};

• the induced outer automorphism of Γ_{∞} is the same as ρ_{∞}τ_{∗}ρ_{∞}^{−1}.

Now, combination of Theorems 15.16, 15.17, 9.24 implies that f_{∞} is Moebius and we extend Γ_{∞} via f_{∞}. The group H = ⟨Γ_{∞}, f_{∞}⟩ is isomorphic to π_{1}(M). Therefore, by Waldhausen-Johannson theorem, the manifolds M and ℍ^{3}/H are homeomorphic. This finishes the 1-st proof in the fibered case.

16 Reduction of the Generic Case of the Hyperbolization Theorem to the Bounded Image Theorem

Step 0. See Sections 19.4–19.6.

16.1 Step 1. Maskit Combination

Theorems 15.5, 15.7 directly follow from the Maskit Combination Theorems 6.68, 6.70. □

16.2 Step 2. Formulation of the Fixed–Point Theorem

Now we shall give yet another reformulation of the generic case of Theorem 15.1, using language of the Teichmüller theory. I recall that T_{Σ}(G) ⊂ T (F_{1}) × T (F_{2}), Section 9.9.

Let c_{j} denote the projections from T (G) to T (F_{j}) (j = 1, 2). The gluing homeomorphism τ of Σ ⊂ ∂_{0}N reverses the induced orientation of the boundary. Consider the product manifold \overline{ℍ^{3}/F_{1}} ∪ \overline{ℍ^{3}/F_{2}} = [−1, 1] × Σ, where we identify \{+1\} × Σ with Ω_{1}/F_{1} ∪ Ω_{2}/F_{2}; Ω_{j} is contained in the domain of discontinuity of Ω(G_{j}) if N is not connected, and in Ω(G) if N is connected.

We extend τ to [−1, 1] × Σ as τ̄(t, z) = (−t, τ(z)). Hence

τ̄ : [−1, 1] × ∂N → ∂N → [−1, 1]

preserves orientation and induces a map of the Teichmüller space τ̄_{∗} : T (F_{1}) × T (F_{2}) → T (F_{2}) × T (F_{1}). Therefore, we can reformulate Theorem 15.1 (Case A) as follows.

Theorem 16.1 There exists a point [ρ] ∈ T_{Σ}(G) such that

τ̄_{∗}(c_{1}([ρ])) = c_{2}([ρ])  (42)

Let’s see how Theorem 16.1 implies Theorem 15.1 (Case A). Let me consider only the case of connected manifold N, the disconnected case is similar. Suppose that there exists a representation ρ as above and ρ(G) = Γ, ρ(F_{j}) = F′_{j}. Then the equality (42) implies that there exists an orientation–preserving isometry α : ℍ^{3}/F′_{1} → ℍ^{3}/F′_{2} which is properly homotopic to the homeomorphism

τ̄ : ℍ^{3}/F′_{1} → ℍ^{3}/F′_{2}

(Here we use the fact that any 1-quasiconformal homeomorphism of \overline{ℂ} is a Moebius transformation.) The isometry α lifts to a Moebius transformation A in the hyperbolic 3-space so that we have:

(a) A conjugates F′_{1} to F′_{2} and the conjugation induces the isomorphism ϕ : F_{1} → F_{2};

(b) since τ flips the boundary components of ∂N × [−1, 1], the transformation A must send Ω′_{1} to \overline{Ω}′_{2}.

Therefore we are in the situation of the 2-nd Maskit Combination Theorem and the discrete group ⟨Γ, A⟩ will uniformize the manifold M = N/τ.

We have the embedding

T (Σ) ⟶^{β} T_{Σ}(G) → T (Σ) × T (\overline{Σ})

Define a map s̄_{G} : T (Σ) → T (Σ̄) as the composition π^{−} ∘ β, where π^{−} is the projection on the second factor in T (Σ) × T (\overline{Σ}). The map s̄_{G} = (s̄_{1}, s̄_{2}) is called the “skinning map” associated with the group G. On the other hand, the gluing involution τ : Σ → Σ induces a map τ̄_{∗} : T (Σ̄) → T (Σ), which is an isometry of Teichmüller spaces. Recall that τ = (τ_{1}, τ_{1}^{−1}) : Σ_{1} ∪ Σ_{2} → Σ_{2} ∪ Σ_{1}.

Theorem 16.2 The following two conditions are equivalent:

(C1) τ̄_{∗}(c_{1}([ρ])) = c_{2}([ρ]);

(C2) τ̄_{∗} ∘ s̄_{G}([ρ]) = [ρ].

Proof: Let [ρ] = α = (α_{1}, α_{2}) ∈ T (Σ_{1}) × T (Σ_{2}) = T (Σ) via the Bers map. Consider (C1). Via Bers identification we have

c_{1}(α) = (α_{1}, s̄_{1}(α_{1}, α_{2}))  (43)

therefore

τ̄_{1}_{∗}(c_{1}(α)) = τ̄_{1}_{∗}(α_{1}, s̄_{1}(α_{1}, α_{2})) = (τ_{1∗}s̄_{1}(α_{1}, α_{2}), τ_{1∗}(α_{1}))  (44)

Then

c_{2}(α_{1}, α_{2}) = (α_{2}, s̄_{2}(α_{1}, α_{2}))  (45)

Thus (C1) is equivalent to

τ_{1∗}s̄_{1}(α_{1}, α_{2}) = α_{2} and τ_{1∗}(α_{1}) = s̄_{2}(α_{1}, α_{2})  (46)

Now consider (C2).

τ̄_{∗} ∘ s̄(α_{1}, α_{2}) = τ̄_{∗}(s̄_{1}(α_{1}, α_{2}), s̄_{2}(α_{1}, α_{2})) =
= (τ_{1∗}^{−1}s̄_{2}(α_{1}, α_{2}), τ_{1∗}s̄_{1}(α_{1}, α_{2}) = (α_{1}, α_{2})  (47)

by (C2). Therefore we get the same system of equations as in (46). □

16.3 Step 3. Proof of the Contraction Theorem

The goal of this step is to prove that the map σ = τ̄_{∗} ∘ s̄_{G} : T (Σ) → T (Σ) is a contraction of the Teichmüller metric d_{T}. The transformation τ̄_{∗} is just an isometry of the Teichmüller spaces T (F_{1}) and T (F_{2}). Therefore, we have to prove that the skinning map s̄ is a contraction. We shall consider only the case when N is connected, the proof in the second case is similar. Let X = (X_{1}, X_{2}), Y = (Y_{1}, Y_{2}) ∈ T (G) = T (Σ_{1}) × T (Σ_{2}) be two distinct points corresponding to Kleinian groups G, Γ. We use the notation V = V_{1} ∪ V_{2}, U = U_{1} ∪ U_{2} for the lifts of Σ to the discontinuity domains Ω(G), Ω(Γ). Each U, V consists of a disjoint union of two open topological disks stabilized by quasifuchsian subgroups F_{j} ⊂ G, F′_{j} ⊂ Γ.

Then the distance between these two points is calculated as follows. Let f_{j} : X_{j} → Y_{j} be the extremal Teichmüller maps which calculate distances between X_{1}, Y_{1} and X_{2}, Y_{2}. Then d_{T} (X, Y) = 1/2 log K, where K = K(f_{1} ∪ f_{2}) ≠ 1.

Now there are two ways to argue, first we will give a proof assuming that Riemann surfaces under consideration are compact.

The quasiconformal map f = f_{1} ∪ f_{2} is not conformal, therefore it has at least one “singular point” (i.e. a point where it is not a diffeomorphism). Lift f to a quasiconformal map f̃ : V → U. Then f̃ extends to a quasiconformal homeomorphism h : \overline{ℂ} → \overline{ℂ} which is conformal on components Ω(G)\G·V and induces the isomorphism G → Γ. This extension has the same coefficient of quasiconformality K as f.

Note that each [Ω(G)−V_{j}]/F_{j} has infinitely many components, therefore the restriction of h to Ω(G) − V_{j} has infinitely many F_{j}–orbits of “singular points”. Thus the projection ĥ of h to a quasiconformal homeomorphism between Σ̄_{j} = [\overline{ℂ} − cl(V_{j})]/F_{j} and Σ̄′_{j} = [\overline{ℂ} − cl(U_{j})]/F′_{j} is not smooth in infinitely many points and, hence, is not a Teichmüller mapping. It implies that ĥ is not extremal in its homotopy class and the extremal map has smaller coefficient of quasiconformality than K.

This argument fails in the case of surfaces with punctures and we need a different argument to prove that ĥ is not a diffeomorphism in a continuum of points. Note that the domain \overline{ℂ} − cl(V_{j}) contains continuum of conical limit points of the group G; Theorem 9.20 implies that h is not differentiable in each of these points. It follows that ĥ is not a Teichmüller mapping.

In any case we conclude that

d_{j} = d_{T} ([Σ̄_{j}, id], [Σ̄′_{j}, ĥ]) < log(K)/2

However the maximum of the distances d_{1}, d_{2} is exactly d_{T} (s̄(X), s̄(Y)). So, the skinning map s̄ is a contraction of the Teichmüller distance. □

From our calculations above we can actually extract more information. Pick a base-point X ∈ T (Σ). Consider a sequence of admissible isomorphisms ρ_{m} : G → Γ_{m} ⊂ P SL(2, ℂ) which corresponds to the sequence of iterations X_{m} := (σ)^{m}(X) ∈ T (Σ). Let

ψ_{m} := ρ_{m}|_{π_{1}(Σ)}

For each m this pair of isomorphisms defines a marking on the Riemann surface Ω(Γ_{m})/Γ_{m}. Thus, X_{m} = [Ω(Γ_{m})/Γ_{m}, ψ_{m}]. For each m we have an isomorphism (ϕ_{m}, ϕ_{m}^{−1}) : π_{1}(X_{m}) → π_{1}(X_{m}) so that

(ϕ_{m}, ϕ_{m}^{−1}) ∘ ψ_{m} = ψ_{m} ∘ τ_{∗}

and (ϕ_{m}, ϕ_{m}^{−1}) maps X_{m} to X_{m+1} as an element of the Teichmüller modular group acting on T (Σ). Our calculations above show that d_{T} (X_{m}, X_{m+1}) < log(K)/2. Hence we can realize each (ϕ_{m}, ϕ_{m}^{−1}) by a K-quasiconformal homeomorphism. Lifting this homeomorphism to Ω(Γ_{m}) and extending it to the limit set we get a sequence of K-quasiconformal homeomorphisms

f_{m} : \overline{ℂ} → \overline{ℂ}, f_{m∗}(Γ_{m}) := f_{m}Γ_{m}f_{m}^{−1} = Γ_{m}

and the restriction of f_{m∗} to the quasifuchsian groups stabilizing components of Ω(Γ_{m}) is the same as τ_{m∗}. Hence by Theorem 9.35 we get the following

Proposition 16.3 For any loxodromic element g ∈ π_{1}(Σ) we have:

K^{−2} ≤ ℓ(ρ_{n}(g))/ℓ(ρ_{n}(τ_{∗}(g)) ≤ K^{2}

where ℓ denotes the translational length.

16.4 Step 4. End of the proof of the Fixed-Point Theorem

Suppose for a moment that we already have proved that the sequence σ^{n}(X) is bounded in the locally compact metric space T_{Σ}(G). Let L denote the closure of the sequence σ^{n}(X). Clearly L is compact and σ : L → L. Let L_{∞} = ∩_{n}σ^{n}(L). Suppose that the mapping σ : L → L does not have a fixed point. Then the continuous function d(σ(x), x) has a positive minimum on L realized by a point x ∈ L, where d is the Teichmüller metric. However the contraction property of σ implies that d(σ^{2}(x), σ(x)) < d(σ(x), x). This contradiction shows that σ has a fixed point. □

17 The Bounded Image Theorem

17.1 Limit exists

I will assume that the manifold N is connected, the other case is absolutely similar (with exception of few notational differences).

Proof of Theorem 15.14.

Recall that all the representations ρ_{n} ∈ D(G, P SL(2, ℂ)), that we have constructed in the previous Chapter, are induced by quasiconformal conjugations of a single Kleinian group G (since we assume that N is connected). Each element in π_{1}(Q_{j}) is parabolic (for every component Q_{j} ⊂ Q), thus

[ρ_{n}] ∈ D(G, π_{1}(Q); P SL(2, ℂ)), ρ_{n} : G → Γ_{n}

where G = π_{1}(N) ⊂ P SL(2, ℂ). Every component of the boundary ∂_{0}N := ∂N − Q is incompressible; if the paired manifold (N, Q) would be acylindrical as well, then the Compactness Theorem 14.92 implies that the sequence [ρ_{n}] is subconvergent in D(G, P SL(2, ℂ)) since the space D(G, π_{1}(Q); P SL(2, ℂ)) is compact.

Assume that the sequence β(σ^{n}(X)) = [ρ_{n}] ∈ T_{Σ}(G) is not relatively compact in D(G, P SL(2, ℂ)). Therefore, we can extract from it a subsequence ρ_{n_{k}} which is convergent to a parabolic action of G on an tree T. We shall establish contradiction with Theorem 11.13 by proving that the action of G on T has a global fixed point. We retain the notation ρ_{n} for ρ_{n_{k}}.

I recall that (N, Q) has a submanifold W which is called “the window” (see Section 2.8). The submanifold W has codimension zero and is a disjoint union of interval bundles over surfaces. Therefore the boundary of W consists of two parts: ∂_{2}W ⊂ ∂N and ∂_{1}W = cl(∂W − ∂_{2}W), the last is the union of “vertical” cylinders.

For each component of ∂_{0}N the action of its fundamental group on T is small and parabolic; the action need not to be minimal, so we will have to take minimal invariant subtrees in T. Then using Theorem 12.29 we define a “maximal elliptic” subsurface B ⊂ ∂_{0}N (with respect to the action of the fundamental groups of components of ∂_{0}N on T). The surface B contains (up to conjugation) all elements g ∈ π_{1}(∂_{0}N) which have fixed points on the tree T. In particular the subsurface B entirely contains each of those components of ∂_{0}N whose fundamental group has a global fixed points on T. According to Theorem 14.1, for each g ∈ π_{1}(∂_{1}(W)) the translational length ℓ_{T} (g) is zero and g has a fixed point on T.

We know from Proposition 16.3, that for every representation ρ_{n} as above and for any g ∈ π_{1}(Σ) = π_{1}(Σ_{1}) ∪ π_{1}(Σ_{2}) ⊂ G we have:

K^{−2} ≤ ℓ(ρ_{n}(g))/ℓ(ρ_{n}(τ_{∗}(g)) ≤ K^{2} or ℓ(ρ_{n}(g)) = ℓ(ρ_{n}(τ_{∗}(g)) = 0

Therefore ℓ_{T} (g) = 0 iff ℓ_{T} (τ_{∗}g) = 0.

If Σ′′′ is a component of ∂_{0}N − Σ, then the marked conformal structure on this surface does not change under quasiconformal deformations in T_{Σ}(G), thus (according to Theorem 6.37) its fundamental group has a global fixed point on T, hence Σ′′′ ⊂ B.

This implies that the surface B must be invariant (up to isotopy) under the involution τ : Σ → Σ, after composing τ with an isotopy of Σ we can assume that τ(B) = B.

Proposition 17.1 B = ∂_{0}N.

Proof: We define an operation π of “pushing through the window” on ∂_{2}W = ∂W ∩ ∂_{0}N as follows. Suppose that γ ⊂ ∂_{2}W is simple loop. If it lies on the boundary of a product component W_{i} = S_{i} × I of W, then take the properly embedded product cylinder C_{γ} ⊂ W_{i}, one of its boundary components is γ, the other one is π(γ) ⊂ ∂_{2}W_{i}. If W_{i} is a nontrivial interval bundle we take C_{γ} to be an essential Moebius band with the boundary component γ; let π(γ) := γ. The cylinder (or Moebius band) C_{γ} is the trace of pushing. Clearly the loops γ, π(γ) are freely homotopic in N, thus the surface B is invariant under the “pushing”.

Suppose that B ≠ ∂_{0}N; then ∂_{0}B = ∂B − Q ⊂ ∂_{2}W (according to Theorem 14.1), this finite collection of loops is invariant under both pushing and τ. Note that if we start pushing a loop in ∂_{0}B, then we never get a loop on ∂_{0}N − Σ since ∂_{0}N − Σ ⊂ B and B is pushing-invariant.

Assume first that none of the components of ∂_{0}B bounds an essential Moebius band in W. Then take a loop γ_{0} ⊂ ∂_{0}B and apply to it the alternating sequence of τ and “pushing” π:

γ_{k} = \underbrace{π ∘ τ ∘ ... ∘ π ∘ τ}_{k times}(γ_{0})

Hence for some k > 0 we get: γ_{k} = γ_{i}, where i ≠ k. The union of traces of pushing of γ_{i}, γ_{i+1}, ..., γ_{k−1} is a collection of annuli. This collection projects to an incompressible torus in N/τ which is not isotopic to any boundary component of N/τ. This contradicts the assumption that N/τ is atoroidal. Now consider the case when a loop γ_{0} ⊂ ∂_{0}B bounds an essential Moebius band C_{γ_{0}} ⊂ W. We apply an alternating sequence of τ and pushing to this loop as above. Then either some γ_{k} (k > 0) again bounds an essential Moebius band C_{γ_{k}} ⊂ W, or before meeting such loop we get a cycle γ_{i} = γ_{k} as above. In the latter case we repeat our argument above. In the former case we take the union of cylinders and Moebius bands

C_{γ_{0}} ∪ ... ∪ C_{γ_{k}}

The image of this union in the manifold N/τ bounds a Klein bottle which is not isotopic to a boundary component. Contradiction. □

Let H_{j} be the images of the fundamental groups of components of ∂_{0}N and of Q in G = π_{1}(N). Each of these groups has a global fixed point on T. Suppose that T is nontrivial. Thus, applying again Corollary 14.74 we conclude that the fundamental group G splits as an amalgamated free product or HNN-extension over a trivial or infinite cyclic group G = A ∗_{Z} B (or G = A∗_{Z}), where each H_{j} is conjugate either to A or to B. Since ∂_{0}N is incompressible we conclude that Z ≅ ℤ. This decomposition must be realizable by a splitting of N along a properly embedded essential annulus or Moebius band C whose fundamental group is Z and ∂C ⊂ ∂N. Hence (up to isotopy) either ∂C ⊂ ∂∂_{0}N, in which case C is not essential (by the definition of the window W), or ∂C ⊂ ∂_{0}N and is not peripheral on this surface, which contradicts the fact that each group H_{j} has a fixed point on T. This implies that there is a fixed point for the action of G on T, which contradicts Theorem 11.13. □

17.2 Geometry of the limit

17.2.1 Preliminaries and notations

Before going into the proof we have to reorganize notations for objects that will appear along the way. Start with the geometrically finite Kleinian group G. We have the quotient manifold M_{0} := M(G) = ℍ^{3}/G, the surface Ω(G)/G which contains a subsurface Σ. From now on I will use the notation Σ′, Σ′′ for the components of this subsurface and τ : Σ′ → Σ′′, τ : Σ′′ → Σ′ for the “gluing involution”. Inside M(G) we have a compact core– the manifold N. We can think about N as the complement to Margulis cusps in the convex manifold C(M(G)). The partial boundary ∂_{0}N contains two incompressible surfaces S′, S′′ (which correspond to Σ′, Σ′′ under the retraction Ω(G) → ∂CΛ(G) ). Each surface Σ′, Σ′′ is the quotient Ω′/F′, Ω′′/F′′, where Ω′, Ω′′ are simply-connected components of Ω(G), and the subgroups F′, F′′ ⊂ G stabilizing these components are quasifuchsian. The surface Ω(G)/G can contain more components, the notation that we reserve for them is: Σ′′′ = Ω′′′/F′′′, the groups F′′′ are quasifuchsian as well. The corresponding component of ∂_{0}N will be denoted S′′′.

The rest of the boundary Q := ∂_{P} N is a collection of annuli and tori. Let M̂′ := ℍ^{3}/F′, M̂′′ := ℍ^{3}/F′′, these are manifolds covering M_{0} = M(G). Let p = (p′, p′′) : M̂ = M̂′ ∪ M̂′′ → M(G) denote the covering map. Each of the manifolds M̂′, M̂′′ topologically is the product Σ′ × ℝ, Σ′′ × ℝ. This identification is chosen so that Margulis cusps in M̂′, M̂′′ correspond to punctured disks in Σ multiplied by the ℝ–factor. Thus, both manifolds M̂′, M̂′′ have exactly two geometrically finite ends: Ê′_{+}, Ê′_{−}, Ê′′_{+}, Ê′′_{−}. The difference between “positive” and “negative” ends is that “positive” ends 1-1 cover actual ends of the manifold M(G) (which are product regions bounded by the surfaces S′, Σ′ and S′′, Σ′′).

We have an isomorphism ϕ : F′ → F′′ (coming from the gluing map τ), it is induced by a homeomorphism f̃: \overline{ℍ^{3}} → \overline{ℍ^{3}} such that K_{O}(f̃) ≤ K^{3} (this is a G-equivariant extension of the K-quasiconformal homeomorphism h : \overline{ℂ} → \overline{ℂ}, see Theorem 9.18). Note that f̃ also conjugates F′′ to F′ (inducing the involution τ_{∗} : (F′, F′′) → (F′′, F′)). The homeomorphism f̃ projects to a quasiconformal homeomorphism

f̂: M̂ → M̂

The latter has the property that it “switches” negative and positive ends (image of negative is positive and vice-versa).

Now, recall that we have a sequence of representations ρ_{n} : G → Γ_{n} ⊂ P SL(2, ℂ), whose restrictions to F′, F′′ we denote ψ_{n} := (ψ′_{n}, ψ′′_{n}). The isomorphisms ρ_{n} are induced by quasiconformal homeomorphisms h̃_{n} : \overline{ℍ^{3}} → \overline{ℍ^{3}}. Thus h̃_{n} project to quasiconformal homeomorphisms h_{n} : M := M(G) → M(Γ_{n}) = M_{n}. Using these homeomorphisms we “transplant” all the notations that we have for the manifold M(G) (and groups, manifolds, maps etc. related to it) to the manifolds M(Γ_{n}) by adding the subscript n. For instance, for each n we have a K^{3}-quasiconformal homeomorphism f̃_{n} : \overline{ℍ^{3}} → \overline{ℍ^{3}}, which induces the isomorphism ϕ_{n} : F′_{n} → F′′_{n}, so that ϕ_{n} ∘ ψ′_{n} = ψ′′_{n} ∘ ϕ. The manifolds M̂′_{n}, M̂′′_{n} have “positive” and “negative” ends Ê′_{±n}, Ê′′_{±n} which are switched by f̂_{n}. The compact manifolds N_{n} are defined as complements to Margulis cusps in the convex cores of M(Γ_{n}).

The situation becomes more delicate when we pass to the algebraic limit:

lim_{n→∞} ρ_{n} = ρ_{∞} : G → Γ_{∞} ⊂ P SL(2, ℂ)

The main problem is that this limit (apriori) does not have to be strong, thus it is not obvious how to translate the geometric information between M(Γ_{n}) and M(Γ_{∞}). The coefficients of quasiconformality of f̃_{n} are uniformly bounded from above by K^{3}, thus (after taking a subsequence if necessary) we get a limit:

lim_{n→∞} f̃_{n} = f̃_{∞}

By continuity, f̃_{∞} is quasiconformal, K_{O}(f̃_{∞}) ≤ K^{3} and f̃_{∞}Γ_{∞}f̃_{∞}^{−1} = Γ_{∞}. Now we introduce “limiting” notations whenever it is possible and add the subscript ∞ to the old notations. Let (ψ′_{∞}, ψ′′_{∞}) := lim_{n→∞}(ψ′_{n}, ψ′′_{n}); (F′_{∞}, F′′_{∞}) := (ψ′_{∞}(F′), ψ′′_{∞}(F′′)). We let M̂_{∞} = M̂′_{∞} ∪ M̂′′_{∞} = ℍ^{3}/F′_{∞} ∪ ℍ^{3}/F′′_{∞}; ϕ_{∞} := ψ′′_{∞} ∘ ϕ ∘ (ψ′_{∞})^{−1} is an isomorphism between F′_{∞} and F′′_{∞} which is induced by f̃_{∞}.

The isomorphism ρ_{∞} sends parabolic elements to parabolic, however the problem is that ρ_{∞} can have accidental parabolic elements. So, as in Section 9.15, we take Ĉ_{∞} to be the union of those Margulis cusps in M̂_{∞} which correspond to cusps in M̂. Let Â_{∞} denote the rest of Margulis cusps in M̂_{∞}, we call them accidental since they correspond to accidental parabolic elements in the groups F′_{∞}, F′′_{∞}.

How many accidental cusps we may have? According to our calculations in Section 6.20: not more than 4(3g − 4 + m), where g is the genus of S′ and m is the number of its boundary components. Thus we get a collection of 2k ≤ 4(3g − 4 + m) primitive elements γ′_{(±j)}, γ′′_{(±i)} ∈ F′, F′′, which represent simple nonperipheral loops on S′, S′′ and are mapped to accidental parabolic elements via ρ_{∞}. Let Π = Π^{+} ∪ Π^{−} ⊂ F′ ∪ F′′ denote the union of F′ and F′′ -conjugacy classes of these elements. Clearly the involution

τ_{∗} = (ϕ, ϕ^{−1}) : (F′, F) → (F′′, F′)

just permutes the elements of Π.

Recall that the manifolds M_{n} may have other ends besides those corresponding to Ê_{±n}. Namely, the quasiconformal homeomorphisms h̃_{n} conjugating G to Γ_{n} are conformal on components Ω′′′ of Ω(G) corresponding to surfaces Σ′′′ = Ω′′′/F′′′ ⊂ Ω(G)/G − Σ. Let h̃′′′_{n} denote the restriction of h̃_{n} to Ω′′′. Then the conformal mappings h̃′′′_{n} are subconvergent uniformly on compacts to a conformal mapping h̃′′′_{∞} : Ω′′′ → Ω′′′_{∞}. Thus the group F′′′_{∞} = ρ_{∞}(F′′′) acts properly discontinuously on Ω′′′_{∞} and the quotient Σ′′′_{∞} := Ω′′′_{∞}/F′′′_{∞} is naturally biholomorphic to Σ′′′. Let G_{∞} denote the geometric limit of the groups Γ_{n}. The fact that we have convergence of the mappings h̃′′′_{n} implies that Ω′′′ is precisely invariant under the subgroup F′′′ in G_{∞}. Finally, the surface Σ′′′_{∞} projects to a surface S′′′_{∞} ⊂ C(Λ(G_{∞})/G_{∞}) under the nearest-point retraction; the end E′′′_{∞} of the manifold M(G_{∞}) bounded by S′′′_{∞} and the surface at infinity Σ′′′_{∞} is geometrically finite. All cusps in this end correspond to punctures on the surface Σ′′′_{∞} ≅ Σ′′′. In particular, if the groups F′_{∞}, F′′_{∞} have some accidental parabolic elements, then the corresponding Margulis cusps are disjoint from E′′′_{∞} (provided that the Margulis constant is chosen to be sufficiently small).

Thus, the geometry of the ends of M_{n} which correspond to Σ′′′ does not change much as we take the limit. This means that we will concentrate mainly on the ends corresponding to Ê_{±n}.

According to Bonahon’s Theorem 13.13, the manifold M̂_{∞}^{1} := M̂_{∞} − C_{∞} is still homeomorphic to the product S×ℝ, where S = S′∪S′′ ⊂ ∂_{0}N is a surface (may be with boundary) and we assume that S × [0, +∞) corresponds to Ê_{+∞} and S × (−∞, 0] corresponds to Ê_{−∞} under orientation-preserving homeomorphism M̂_{0}^{0} → M̂_{∞}^{1}. Now we can use Lemma 9.40 to construct immersions

λ_{n} : M_{∞}^{c} → M_{n}
λ̂_{n} : M̂_{∞}^{c} → M̂_{n}

where M̂_{∞}^{c}, M_{∞}^{c} are compact cores in M̂_{∞}^{0}, M_{∞}, so that:

• λ̂_{n} is a lift of λ_{n};

• lifts of λ̂_{n} to the universal covers are convergent to the identity uniformly on compacts.

Warning 17.2 Obviously M̂_{∞}^{c} does not cover M_{∞}^{c}, unless N is an interval bundle over a surface.

Identify a compact core M̂_{∞}^{c} with S × [−1, 1]. Assuming that M̂_{∞}^{c} contains S × {0} we get a sequence of maps λ̂_{n} : S → M̂_{n} which are homeomorphic embeddings near the boundary of S. According to our analisys in Section 9.15, we can find properly homotopic embeddings ê_{n} : S → M̂_{n} in the 1-neighborhood of each λ̂_{n} (for sufficiently large n). Thus, the surfaces ê_{n}(S) split M̂_{n} into positive and negative ends. In the end of Section 9.15 we have established a natural 1-1 correspondence between ends of the manifold M̂_{n}^{0} and topological ends of the manifold M̂_{∞}^{1}.

This allows us to assign the names “positive” and “negative” to the ends Ê′_{±∞}, Ê′′_{±∞} of the manifold M̂_{∞}^{1}, so that λ̂_{n} sends positive ends to positive and negative ends to negative in the following sense:

λ̂_{n}(S × {+1}) ⊂ Ê_{+n}, λ̂_{n}(S × {−1}) ⊂ Ê_{−n}

Note that this “assignment” is purely geometric and (apriori) has nothing to do with topology of the manifold M_{∞}^{1}, since its compact core is only homotopy-equivalent to the compact core M_{n}^{c} and does not have to be homeomorphic. This assignment is consistent with the orientation: the surfaces S′, S′′ are canonically oriented as well as the manifold M̂_{∞}^{c}, if the end Ê′_{+} lies to the “right” from S′ ⊂ M̂_{0}^{c}, then the end Ê′_{+∞} lies to the “right” from the corresponding surface S′ × {0} ⊂ M̂_{∞}^{c}.

Nevertheless, the limiting map f̂_{∞} “switches” positive to negative ends and vice-versa because distance between the mappings

λ̂_{n} ∘ f̂_{∞}, f̂_{n} ∘ λ̂_{n}

tends to zero as n → ∞. Accordingly, accidental cusps in Â_{∞} group into “positive cusps” Â_{+∞} = Â′_{+∞} ∪ Â′′_{+∞} which belong to Ê_{+∞} and “negative cusps” Â_{−∞} = Â′_{−∞} ∪ Â′′_{−∞} which belong to Ê_{−∞}. The map f̂_{∞} sends parabolic elements to parabolic and does not mix accidental and regular parabolic elements. Thus (up to isotopy) f̂_{∞} will switch positive to negative cusps. Note that the fundamental group of any “negative” accidental cusp in Ê_{−∞} is not conjugate in π_{1}(M̂_{∞}) to the fundamental group of any positive cusp since they are disjoint (see Lemma 6.48).

What happens in the “pre-limit” hyperbolic manifolds M̂_{n} with these “accidental” cusps? They correspond to Margulis tubes for which we shall use the notation Â_{n} = Â′_{+n} ∪ Â′′_{+n} ∪ Â′_{−n} ∪ Â′′_{−n}.

Inside of the manifold M_{∞}^{1} we have a compact core M_{∞}^{c} of diameter d. Let ν be the injectivity radius at M_{∞}^{c}. Recall that M_{∞}^{1} was constructed given the Margulis constant μ_{3}. Apriori this compact core intersects the union of accidental Margulis cusps A_{∞}. However we can choose a smaller constant ϵ < μ_{3} so that:

• (d + 2)-neighborhood of M_{∞}^{c} is disjoint from A_{∞} ∩ M(Γ_{∞})_{(0,ϵ]};

• μ_{3} > ν > θ(K^{2k}, ϵ), where 2k is the cardinality of Π and θ is the function introduced in Section 4.10.

We shrink Margulis cusps A_{∞} to components A_{∞}∩M(Γ_{∞})_{(0,ϵ]}. To simplify the notations we will retain the notation A_{∞} for the smaller accidental cusps. Similarly we reduce μ_{3} to ϵ for Margulis tubes A_{n} ⊂ M_{n}, which converge to A_{∞}. Again we retain old notation for the new (smaller) Margulis tubes. Thus the 2-neighborhood of λ_{n}(M_{∞}^{c}) is disjoint from A_{n} for all sufficiently large n. Moreover, for each i ≤ 2k we have:

p_{n}(f̂_{n}^{i}(Â_{n})) ∩ Nbd_{2}(λ_{n}(M_{∞}^{c})) = ∅

since K_{O}(f̂_{n}^{i}) ≤ K^{i} and distortion of length by quasiconformal mappings is bounded from above according to the estimates of Section 4.10. Thus p_{n}(f̂_{n}^{i}(Â_{n})) is also disjoint from the embedded surfaces e_{n}(S′), e_{n}(S′′); therefore each of components of p_{n}(f^{i}(Â_{n})) is forced to be contained in one of the ends E′_{+n}, E′′_{+n} bounded by e_{n}(S′), e_{n}(S′′).

Now, at last, we are ready to discuss the idea of the proof.

17.2.2 Idea of the proof

We have to prove that the pair of isomorphisms

(ψ′_{∞}, ψ′′_{∞}) : (F′, F′′) → (F′_{∞}, F′′_{∞})

has no accidental parabolic elements and the groups in the image are geometrically finite.

I will outline the proof assuming that Q = ∅, ∂N = S ≅ Σ and the manifold N is acylindrical. Let M_{∞}^{c} be the Scott’s compact core in the manifold ℍ^{3}/Γ_{∞}. Theorem 2.34 implies that the isomorphism ρ_{∞} : π_{1}(N) → Γ_{∞} = π_{1}(M_{∞}^{c}) is realized by a homeomorphism

h_{∞} : N → M_{∞}^{c}

In particular, h_{∞}(S) = S_{∞} consists of the pair of boundary surfaces of M_{∞}^{c}, whose union is the entire boundary of this manifold.

The manifold M̂_{∞} has four ends: Ê′_{±∞}, Ê′′_{±∞} (see previous section). Since the maps λ̃_{n} are convergent to the identity uniformly on compacts, we conclude that the “positive” ends of M̂_{∞} 1-1 cover ends of the manifold M_{∞} = M(Γ_{∞}). The quasiconformal homeomorphism f̂_{∞} switches positive and negative ends of M̂_{∞}.

Suppose that M̂′_{∞} has cusps on the positive side, thus M̂′′_{∞} must have cusps on the negative side as well. Suppose that such cusp Ĉ^{−} ⊂ Ê_{−∞} ⊂ M̂′′_{∞} projects to a cusp in the end of M_{∞} bounded by S′_{∞}. Then π_{1}(Ĉ^{−}) ⊂ π_{1}(S′′_{∞}) = F′′_{∞} is conjugate to an infinite cyclic subgroup of π_{1}(S′_{∞}) = F′_{∞}, which is impossible since N is acylindrical. Suppose that Ĉ^{−} projects to a cusp C^{−} in the “positive” end E′′_{+∞} of M_{∞} bounded by S′′_{∞}. Recall that the covering Ê′′_{+∞} → E′′_{+∞} is 1-1 and, hence, we can lift C^{−} to a cusp Ĉ^{+} in the positive end Ê′′_{+∞}. The cyclic groups π_{1}(Ĉ^{+}), π_{1}(Ĉ^{−}) ⊂ F′′_{∞} can not be conjugate in F′′_{∞} since these cusps are disjoint (see Lemma 6.48). However, they both project to C^{+}, hence these cyclic groups are conjugate in Γ_{∞} which contradicts our assumption that N is acylindrical.

But where the projection C^{−} can lie in M_{∞}? It can not lie in any of two ends and it can not lie in the compact M_{∞}^{c}. Therefore we are left with only one possibility: there are no cusps on the positive side of M̂_{∞}. However the same argument also shows that there are no cusps on the negative side of M̂_{∞} as well. So we conclude that the groups F′_{∞}, F′′_{∞} have no parabolic elements at all !

The next stage of the proof is to establish that the groups F′_{∞}, F′′_{∞} are geometrically finite.

Theorem 13.19 immediately implies that the negative ends Ê′_{−∞}, Ê′′_{−∞} can not be geometrically infinite since the fundamental group of N is not a finite extension of the surface group. The limiting quasiconformal homeomorphism f̂_{∞} switches positive and negative ends Ê_{±∞}. This implies that all ends of M̂_{∞} are geometrically finite. Thus F′_{∞}, F′′_{∞} are both geometrically finite and have no parabolic elements. Therefore they are quasifuchsian groups and hence must belong to T (F′), T (F′′).

We conclude that the sequence σ^{n}(X) = β^{−1}ρ_{n} is convergent in T (Σ), hence we are done.

17.2.3 Limit has no accidental parabolics

Theorem 17.3 The subgroups F′_{∞}, F′′_{∞} of Γ_{∞} have no accidental parabolic elements.

Proof: Suppose that the set of accidental parabolic loops Π is not empty. To simplify the discussion let me first assume that Π consists exactly of one positive loop [γ_{+}] ∈ Π′_{+} and one negative loop [γ_{−}] ∈ Π′′_{−}, where γ_{±} denotes an element of F′, F′′ which represents the corresponding loop. Since τ_{∗} switches Π_{+} and Π_{−}, we conclude that τ_{∗}(γ_{+}) = γ_{−} and τ_{∗}(γ_{−}) = γ_{+}. Start with the Margulis tube Â_{+n} ⊂ M̂′_{n} which corresponds to γ_{+}, the image p_{n}(f̂_{n}Â_{+n}) is contained either in E′_{+n} or in E′′_{+n} (see our discussion in Section 17.2.1). In any case there exists an isometric lift of p_{n}(f̂_{n}Â_{+n}) to Ê_{+n}, which must belong to the θ(K^{3}, ϵ)-thin part of Ê_{+n}, i.e. is contained in one of the Margulis tubes B̂_{+n}. Algebraically this means the following. The element τ_{∗}(γ_{+}) = γ_{−} is conjugate to an element α_{+} which belongs either to F′ or to F′′ (depending on the end E′_{+n}, E′′_{+n} where p_{n}(f̂_{n}Â_{+n}) is). This element is nonperipheral in F′, F′′. Let me consider first the case α_{+} ∈ F′ (see Figure 53)). The F′-conjugacy class of α_{+} depends only on topology on N and does not depend on the number n at all. So, let me choose α_{+} within its conjugacy class once and for all (independently on n). The translational length ℓ(ρ_{n}(α_{+})) ≠ 0 is at most K^{2}ℓ(ρ_{n}(γ_{+})), hence it tends to zero as n → ∞. Therefore α_{+} is an accidental parabolic element (ρ_{∞}(α_{+}) is parabolic). However we have a complete list of (conjugacy classes) of accidental parabolic elements: we conclude that α_{+} = (γ_{+})^{s} (up to conjugation in F′). The elements γ_{+}, γ_{−} are primitive in the groups F′, F′′. Thus Lemma 2.52 implies that s = ±1.

Figure 53: Projections of Margulis tubes.

Recall that to get a presentation for the HNN extension π_{1}(N/τ) of G = π_{1}(N), we add to the list of generators in G a letter t and add relations:

tF′t^{−1} = F′′, tgt^{−1} = ϕ(g), g ∈ F′

Therefore in the group π_{1}(N/τ) we get a trivial Baumslag-Solitar relation

tγ_{+}t^{−1} = (γ_{+})^{s}

It means that we have got a nonperipheral subgroup Z × ℤ ⊂ π_{1}(N/τ) (which has finite index in the subgroup generated by t, γ_{+}), this contradicts the assumption that N/τ is atoroidal.

Our argument in the case α_{+} ∈ F′′ is similar. After conjugation in F′′ the element α^{+} equals (γ_{−})^{s}. However the Margulis tube corresponding to γ_{−} is contained in the “negative” part of M̂′′_{n}, meanwhile the Margulis tube B̂_{+n} corresponding to α^{+} must be contained in the “positive part”. This is impossible.

Now we consider the general case (when the cardinality of Π does not have to be 2). Let Π_{+} = {γ_{+1}, ..., γ_{+k}}, Π_{−} = {γ_{−1}, ..., γ_{−k}}. The argument remains pretty much the same. We start with the Margulis tube Â_{+n}^{1} ⊂ M̂′_{n} which corresponds to γ_{+1}. Then p_{n}(f̂_{n}Â_{+n}^{1}) is contained either in E′_{+n} or in E′′_{+n}. Suppose that it is contained in E′_{+n}. We lift it isometrically into one of the Margulis tubes B̂_{+n}^{1} which is contained in the θ(K^{3}, ϵ)-thin part of Ê′_{+n}. Algebraically this corresponds to the fact that γ_{−1} = τ_{∗}(γ_{+1}) is conjugate in G to an accidental parabolic element α_{+1} = (γ_{+i})^{s_{1}}, where γ_{+i} ∈ F′.

Figure 54: Construction of a torus or a Klein bottle.

Both loops [α_{+1}], [γ_{−1}] are simple, they are homotopic in N but not in ∂_{0}N, so we connect them by an essential annulus L_{1} ⊂ N. If i = 1 then we are done as above. Otherwise we let i = 2 and apply the same procedure as before to the element γ_{+2}, etc. On each step j ≤ k of induction, p_{n}(f̂_{n}B̂_{+n}^{j}) is contained in the θ(K^{3j}, ϵ)-thin part of M_{n}, therefore it is in one of the product ends E′_{+n}, E′′_{+n} and can be isometrically lifted to Ê_{+n}. After at most k iterations we get stuck: α_{+i} = γ_{+j}. This means that inside of the manifold N/τ we have constructed an incompressible nonperipheral torus or Klein bottle (see Figure 54); which contradicts the assumption that N/τ is atoroidal. The only way out is that ψ′_{∞}, ψ′′_{∞} do not have accidental parabolic elements. □

17.2.4 Endgame: Limit is geometrically finite

So, the groups F′_{∞}, F′′_{∞} do not have accidental parabolic elements. We know that the limiting quasiconformal homeomorphism f̃_{∞} conjugates F′_{∞} to F′′_{∞}. Suppose that one of the limiting groups (say F′_{∞}) is not geometrically finite. Then the group F′′_{∞} is geometrically infinite as well. According to Theorem 13.19 it implies that the manifold M_{∞}^{0} has exactly two geometrically infinite (geometrically tame) ends E′_{+∞}, E′′_{+∞} and the subgroups F′_{∞}, F′′_{∞} are peripheral in Γ_{∞} ≅ π_{1}(M_{∞}^{0}). (See Figure 55).

Proposition 17.4 The convergence ρ_{n} → ρ_{∞} is strong.

Proof: Let G_{∞} denote the geometric limit of the sequence of groups Γ_{n}. We have the covering M(Γ_{∞}) → M(G_{∞}) induced by the inclusion Γ_{∞} ↪ G_{∞}. Again, according to Theorem 13.19 we conclude that the restriction of this covering to the geometrically infinite ends E_{+∞} of M_{∞}^{0} is finite-to-1, the restriction of the covering to the other (geometrically finite) ends is 1-1 as well (see our discussion in Section 17.2.1).

Hence the whole covering M(Γ_{∞}) → M(G_{∞}) is finite-to-1. This implies that Γ_{∞} has finite index in G_{∞}, therefore by Theorem 9.12 these groups are equal and the convergence is strong. □

Figure 55: Ends of the limiting manifold.

Now there are several ways to argue. Here is one. Lemma 9.41 implies that the maps λ_{n} are injective for large n, therefore they give homeomorphisms between compact cores of the manifolds M_{∞}^{0} and M_{n}^{0} for large n. Hence for the ends Ê_{+∞} of M̂_{∞} which were assigned the “positive” labels in the Section 17.2.1, the covering p_{∞} : Ê_{+∞} → M_{∞} is indeed injective. Thus the limiting quasiconformal mapping f̂_{∞} sends Ê_{+∞} to Ê_{−∞}. We have concluded that both ends Ê′_{+∞}, Ê′′_{+∞} are geometrically infinite. Therefore both ends Ê′_{−∞}, Ê′′_{−∞} are geometrically infinite as well, which contradicts Theorem 13.19. Hence both groups F′_{∞}, F′′_{∞} are geometrically finite. This finishes the proof of the Bounded Image Theorem. □

18 Hyperbolization of Manifolds Fibered Over 𝕊^{1}

18.1 Compactness theorem for aperiodic homeomorphisms

Let S be a Riemann surface of finite hyperbolic type, Γ = π_{1}(S). We assume Γ ⊂ P SL(2, ℝ) is a Fuchsian group so that S = ℍ^{2}/Γ. Recall that S̄ = (\overline{ℂ}−cl(ℍ^{2}))/Γ has the same marked hyperbolic structure as S and the opposite orientation. These surfaces are identified via projection of the complex conjugation z ↦ z̄ to a homeomorphism S → S̄. There is a continuous length function

ℓ : T (S) × ML(S) → ℝ

which extends the length function for closed geodesics, see Section 12.16. Suppose that

(ψ_{n}^{+}, ψ_{n}^{−}) ∈ T (S) × T (S̄)

is a sequence and (λ^{+}, λ^{−}) is a pair of elements in ML(S) × ML(S̄) so that

ℓ(ψ_{n}^{±}, λ^{±}) ≤ L < ∞

Consider the sequence [ρ_{n}] ∈ T (Γ) which corresponds to (ψ_{n}^{+}, ψ_{n}^{−}) under the Bers isomorphism.

Theorem 18.1 Double limit theorem (W. Thurston, J.- P. Otal.) Suppose that the pair of measured geodesic laminations λ^{+}, λ^{−} is binding the surface S. Then the sequence [ρ_{n}] is subconvergent in the representation variety

R(Γ, P SL(2, ℂ))

In the next section I will give an outline of the proof, for details read [162], [195].

Now we shall see how Theorem 15.16 can be deduced from the Double Limit Theorem. Suppose that τ : S → S is an aperiodic homeomorphism. Its stable and unstable measured geodesic laminations are λ^{+}, λ^{−} and they are binding the surface S. I recall that there is a constant k > 1 so that

τ(λ^{±}) = k^{±1}λ^{±}

Length of the lamination τ(λ^{±}) in the marked hyperbolic surface ψ_{n}^{±} := τ^{±n}(S) is the same as the length of λ^{±} on S, which we shall denote by L^{±}.

Thus the length of λ^{±} in ψ_{n}^{±} is L^{±}/k^{n}, which tends to zero as n → ∞. So we are exactly in the situation when we can apply Theorem 18.1. □

18.2 The double limit theorem

Let me start the proof with an example where λ^{+} = α, λ^{−} = β is a pair of binding simple closed geodesics on S. Let P be the parabolic locus of S (i.e. the collection of disjoint Margulis cusps). We will regard α, β as elements of the fundamental group Γ of S. Suppose that the sequence of conjugacy classes of representations [ρ_{n}] ∈ D(Γ; P; SL(2, ℂ)) is not precompact in the representation variety. Then ρ_{n} is subconvergent to a minimal, small, parabolic and nontrivial action of the group Γ on a tree T. Theorem 6.37 tells us that translational lengths ℓ(ρ_{n}(α)) and ℓ(ρ_{n}(β)) are uniformly bounded from above by 2L. However, according to construction of the tree T, this implies that both α and β have fixed points on T:

ℓ_{T} (α) = lim_{n→∞} ℓ(ρ_{n}(α))/D_{n} = 0, ℓ_{T} (β) = 0

since D_{n} → ∞. Now, it follows from Theorem 12.29 that there is a (maximal elliptic) subsurface B ⊂ S which contains homotopy classes of all loops whose representatives in π_{1}(S) = Γ have fixed points on T. Our assumption that α, β are binding implies that S = B and therefore Γ has a global fixed point on T. Contradiction. □

Now let us consider the general case. Suppose that the sequence of conjugacy classes of representations [ρ_{n}] is not precompact in the representation variety. Recall that for each representation ρ_{n} : Γ → P SL(2, ℂ) we had a number D_{n} = D_{ρ_{n}} such that after rescaling ℍ^{3} by D_{ρ_{n}}^{−1}, the sequence of representations ρ_{n} is convergent to an action of the group Γ on a tree T. We have identified the space of projective classes of laminations PML(S) with a “unit sphere” Σ_{S} in ML(S) (consisting of normalized laminations μ/p(μ), see Section 12.15). The subset C(S) of simple closed geodesics on S is embedded into Σ_{S} as: γ ↦ γ/p(γ). Let Σ_{S}^{0} denote the image. As we know Σ_{S}^{0} is dense in Σ_{S}. Note that the translational length under representations ρ : Γ → Isom(ℍ^{3}) defines a function

ℓ : Σ_{S}^{0} × D(Γ, P; P SL(2, ℂ)) → ℝ

by the formula

ℓ(γ/p(γ), ρ) = ℓ(ρ(γ))/[p(γ) · D_{ρ}]

It turns out that there exists a continuous extension of this function to

ℓ : Σ_{S} × D(Γ, P; P SL(2, ℂ)) → ℝ

Recall that T (Γ, P) is the space of small, minimal and parabolic actions of the group Γ on metric trees. Each action is uniquely determined by the corresponding translational length function ℓ_{T} : Γ → ℝ. The space T (Γ; P) has the topology of convergence of the length functions on a finite set of generators of Γ, and it is naturally homeomorphic to ML(S) (see Section 12.11). The space D(Γ, P; P SL(2, ℂ)) has the compactification

D̄(Γ, P; P SL(2, ℂ)) ⊂ D(Γ, P; P SL(2, ℂ)) ∪ T (Γ; P)

(see Section 11.5).

The main technical result of Otal [162] (see also [159]) is the following:

Theorem 18.2 For each σ ∈ Σ_{S} the function ℓ(σ, •) admits a lower semicontinuous extension

ℓ̄(σ, •) : D̄(Γ, P; P SL(2, ℂ)) → ℝ

so that ℓ̄(σ, T_{λ}) = i(σ, λ) is the geometric intersection number.

(Actually Otal’s proof with a little more care implies continuity of this extension as well.)

Now we can deduce the “Double Limit Theorem” from this information. Recall that for each simple closed geodesic γ on the marked hyperbolic surface ψ_{n}^{±} ∈ T (S), the translational length of the ρ(γ) on ℍ^{3} is at most 2Length_{ψ_{n}^{±}}(γ), according to Theorem 6.37. (Here we regard γ as an element of π_{1}(S) = Γ.) On the other hand, multiples of simple closed geodesics are dense in Σ_{S} and the length function ℓ : Σ_{S} × D(Γ, P; P SL(2, ℂ)) → ℝ is continuous. Thus

ℓ(λ^{±}, [ρ_{n}]) ≤ 2L

Hence, after taking a subsequence we get:

lim_{n→∞} D_{n}^{−1}[ρ_{n}(Γ) ↷ ℍ^{3}] = [G ↷ T_{ξ}]

where T = T_{ξ} is a Γ-tree dual to the measured geodesic lamination ξ and D_{n} := D_{ρ_{n}}. Therefore

i(ξ, λ^{±}) ≤ lim_{n→∞} D_{n}^{−1}ℓ(λ^{±}, [ρ_{n}])/p(λ^{±}) ≤ lim_{n→∞} D_{n}^{−1}2L/p(λ^{±}) = 0

This means that i(ξ, λ^{±}) = 0. However, the laminations λ^{+}, λ^{−} are binding the surface S, and we get a contradiction. □

18.3 Proof of Theorem 13.17

We already have proved that the sequence of representations ρ_{n} : F → F_{n} ⊂ P SL(2, ℂ) is subconvergent to a representation

ρ_{∞} : F → F_{∞}

Note that the K-quasiconformal homeomorphism τ : S → S is realized by K-quasiconfor-mal homeomorphisms of the pair of marked Riemann surfaces (τ^{n}(α), τ^{−n}(α)). Recall that we are dealing with a sequence of representation ρ_{n} : F → F_{n}, which correspond to the points (τ^{n}(α), τ^{−n}(α)) ∈ T (S) × T (S̄) under the Bers isomorphism. Thus for each n we get a K-quasiconformal homeomorphism

f_{n} : \overline{ℂ} → \overline{ℂ}

so that

f_{n∗} ∘ ρ_{n} = ρ_{n} ∘ τ_{∗}

where f_{n∗} : F_{n} → F_{n} is an automorphism induced by f_{n} and τ_{∗} : F → F is the automorphism induced by τ. Hence, by the convergence property of sequences of uniformly quasiconformal homeomorphisms (Section 5.1), the sequence f_{n} is subconvergent to a Kquasi-conformal homeomorphism f : \overline{ℂ} → \overline{ℂ} which conjugates F_{∞} to itself and

f_{∞∗} ∘ ρ_{∞} = ρ_{∞} ∘ τ_{∗}

Again, for simplicity of discussion I will consider the case when S is a closed surface. Quasiconformal homeomorphism f_{∞} must conjugate parabolic elements to parabolic. On the other hand, according to Sullivan Finiteness Theorem there are only finitely many conjugacy classes of parabolic elements in F_{∞} and they all correspond to simple closed loops γ_{j}^{±} on S (see Section 6.20 for discussion of these loops). If there is at least one parabolic element in F_{∞}, then τ must permute the isotopy classes of the loops γ_{j}^{±}, hence some iteration of τ keeps all these classes fixed, which contradicts the assumption that τ is aperiodic. Thus the group F_{∞} contains no parabolic elements whatsoever and we are in position to apply Theorem 9.39. If the group F_{∞} has nonempty domain of discontinuity Ω_{∞}^{0}, it must be simply-connected. Hence, according to Theorem 9.39, marked conformal structures on surfaces Ω(F_{n})^{0}/F_{n} are convergent to the marked conformal structure on the surface Ω_{∞}^{0}/F_{∞}. This contradicts the fact that the marked conformal structures τ^{±n}(S) do not accumulate in the Teichmüller space T (S) (and T (S̄)), because of discreteness of the action of the modular Mod_{S} on T (S). Therefore Ω(F_{∞}) is empty. □

18.4 An Alternative Approach to Hyperbolization of Manifolds Fibered Over 𝕊^{1}: Reduction to the Generic Case

In this section we describe an alternative approach to hyperbolization of 3-manifolds fibered over the circle.

Theorem 18.3 Suppose that M is a compact atoroidal 3-manifold fibered over 𝕊^{1} so that the fiber has negative Euler characteristic. Assume that M admits a finite covering M′ such that:

rank H_{2}(M, ∂M; ℤ) ≥ 2

Then M is hyperbolic.

Remark 18.4 It was proven in [41] that if M is a compact 3-manifold fibered over circle, ∂M ≠ ∅ and π_{1}(M) is not virtually solvable then M has a finite cover M′ such that dim(H_{2}(M′, ∂M′; ℝ)) ≥ 2. Conjecturally, the assumption that ∂M ≠ ∅ can be removed.

Proof: We apply Theorem 3.1 to the manifold M′. Note that, according to our discussion in Section 2.9, the only case when an embedded (connected) superincompressible surface S is not a fiber in a fibration of M over 𝕊^{1}, but is a virtual fiber, is: S separates M into two components. Such surface is homologically trivial, thus, the superincompressible surface given by Theorem 3.1 is not a virtual fiber. Therefore we apply the generic case of Thurston’s hyperbolization theorem to the manifold M′ and conclude that M′ is hyperbolic. Thus Theorem 9.22 implies that M is hyperbolic as well. □

19 Hyperbolization of Orbifolds

19.1 Hyperbolic orbifolds associated with patterns of circles

In this section we shall describe a procedure of constructing hyperbolic orbifolds and discrete subgroups of Isom(ℍ^{3}) associated with patterns of round discs on Riemann surfaces. This construction will be used repeatedly in this Chapter.

Suppose that G ⊂ Isom(ℍ^{3}) is a geometrically finite torsion-free Kleinian group, S(G) = Ω(G)/G is the quotient surface for the domain of discontinuity. We define round disks in S(G) as follows. First of all, we count as round disks homeomorphic projections of compact round disks from Ω(G) to S(G). The second class of “round disks” consists of closed punctured disks D in S(G), where the puncture corresponds to a cusp and the disk D is covered by a round disk Δ−{p} ⊂ Ω(G), so that Δ is precisely invariant under its parabolic stabilizer A in G and p is the fixed point of A.

Suppose that P = D_{1} ∪ ... ∪ D_{n} is a finite collection of closed round disks in S(G). For each disk we have an embedded hyperbolic half-space in M(G) = ℍ^{3}/G, namely the convex hull C(D_{i}) of D_{i} in M(G). Let C(P) := ∪_{i=1}^{r}C(D_{i}) and M^{∗} := M(G) − int(C(P)). The boundary of the manifold M^{∗} has a natural polyhedral piecewise-geodesic structure (faces are pieces of ∂C(D_{i})’s, etc.). For any two adjacent faces F_{i}, F_{j} ⊂ ∂M^{∗} we have the angle θ_{ij}. We assume that this polyhedral structure gives M^{∗} a structure of orbifold. This means that:

1. The angles θ_{ij} belong to {π/n; n = 2, 3, ...}.

2. There is no vertices which belong to more than 3 faces.

3. If v is a vertex, then the angles α, β, γ between the edges emanating from v satisfy

the inequality

α + β + λ > π

If these conditions are satisfied we will say that P is a pattern of round disks in S(G). Let O denote the resulting locally reflective orbifold with the underlying space M^{∗}.

Our goal is to give this orbifold a canonical complete hyperbolic structure and describe the holonomy group of this structure.

Let G^{∗} ⊂ Isom(ℍ^{3}) be the group generated by G and reflections in all lifts of the hyperplanes ∂C(D_{i}), i = 1, ..., r, into ℍ^{3}. This group is called the group associated with the pattern P.

Figure 56: Case (a) of two tangent discs.

Theorem 19.1 The group G^{∗} is geometrically finite and the orbifold ℍ^{3}/G^{∗} is naturally diffeomorphic to O.

Proof: I will present a proof in the case when the manifold M(G) has no cusps, the general case is similar.

Consider the convex domain Φ ⊂ ℍ^{3} which is the complement to inverse images of the open hyperbolic half-spaces intC(D_{i}), i = 1, ..., r. This is a locally finite G-invariant polyhedron so that Φ/G^{∗} = M^{∗}. Thus, according to Poincare’s Fundamental Polyhedron Theorem 6.22, Φ is a fundamental domain for the group R generated by reflections in faces of Φ. Locally the polyhedron Φ has the same structure as the orbifold O.

Figure 57:

The quotient orbifold ℍ^{3}/R is naturally identified with Φ (where the orbifold structure is given by lift from O). The group G acts on Φ as the group of covering transformations with the quotient M^{∗}. So, it is left to prove that the group G^{∗} generated by R and G is geometrically finite. For this we have to remove a collection of cuspidal neighborhoods of cusps of the orbifold Ō = (ℍ^{3} ∪ Ω(G^{∗}))/G^{∗} so that the complement is compact. Consider a regular neighborhood N ⊂ M̄(G) of the convex hull C(P). Then the manifold M̄(G) − N is naturally homeomorphic to M̄(G). The frontier of N in M̄(G) is disjoint from the thin part of the orbifold O. Recall that the group G was convex-cocompact to begin with. Thus the M̄(G) − N is compact.

Now we need to understand cusps which are contained in the orbifold N. Some of these cusps correspond to isolated ideal vertices of N. Namely, the intersection of the polyhedron N with ∂M̄ can contain isolated points x. Either three or four faces of Ō meet at such points. These points correspond to cusps of virtual rank 2.

The second class of cusps corresponds to ideal vertices of the complement ∂M̄ − P. We call them nonisolated ideal vertices x of N. At such points two disks from the pattern P are tangent.

We have the following combinatorial possibilities for x. First two are cases of nonisolated ideal vertices.

(a) x belongs to exactly two circles ∂D_{1}, ∂D_{2} which are tangent at x. Then we can find two compact round disks Δ_{1}, Δ_{2} ⊂ S(G) which are tangent to each other at the point x and which are orthogonal to ∂D_{1}, ∂D_{2}. (Figure 56.) The union of the convex hulls of Δ_{1}, Δ_{2} with a horoball with center at x forms a cuspidal neighborhood of the cusp of O corresponding to the point x.

(b) The next case is when the point x belongs to 3 circles ∂D_{1}, ∂D_{2}, ∂D_{3} so that ∂D_{1}, ∂D_{3} are mutually tangent and ∂D_{3} is orthogonal to both of them. In this case we take a small round disk Δ which is tangent to D_{2} at the point x, orthogonal of D_{1}, D_{3} and is disjoint from all other disks in P. The union of Δ with a horoball with center at x forms a cuspidal neighborhood of the cusp of O corresponding to the point x. (Figure 57.)

Figure 58: Shrink circles. Blow up the point x into a triangle or a square.

(c) Case of an isolated vertex x. In this case a horoball with center at x is a cuspidal neighborhood for the point x. (On Figure 57 we describe the case when x is a common point of four circles.)

After removing these cuspidal neighborhoods we get a compact orbifold. Thus G^{∗} is geometrically finite. □

We use our analisys of the cusps in O to describe a compactification of O to an orbifold with boundary. I recall that in Section 4.7 we proved that there is no point in S(G) which is covered by more than 3 open disks in P. Thus, given the configuration P we can construct a graph K ⊂ S(G) as follows.

Step 1. We start by decreasing slightly all disks in the family P so that each new disk D′_{i} is disjoint from ∂D_{i}. This procedure keeps combinatorics of P except in the points x where either:

1. Two disks D_{i}, D_{j} are tangent: the new disks D′_{i}, D′_{j} become disjoint near x, instead of x we insert a 2-edged square orbifold P_{ij};

2. Two disks D_{i}, D_{j} are tangent and a disk D_{k} is orthogonal to both of them at the point x; instead of x we insert a 3-edged square orbifold P_{ijk};

3. Four circles ∂D_{i}, ∂D_{j}, ∂D_{k}, ∂D_{r} have the common point x, pairs of disks ∂D_{i}, ∂D_{j} and ∂D_{k}, ∂D_{r} are tangent and D_{i} is orthogonal to D_{k}; instead of x we insert a mirror square orbifold P_{ijkr};

4. Three circles ∂D_{i}, ∂D_{j}, ∂D_{k} have the common point x; after we pull them apart we insert instead of x a Euclidean triangle orbifold P_{ijk}.

See Figures 58, 59.

Step 2. In the new configuration of disks D′_{j} we collapse each “triangle” D_{i} ∩ D_{j} ∩ D_{k} to a point (see Figure 60) and retain notations D′_{j} for disks.

Figure 59: Shrink circles, blow up the point x into a square.

Step 3. Finally we collapse each bigon D′_{i} ∩ D′_{j} to a segment [v_{ij}, v_{ji}] ⊂ D′_{i} ∩ D′_{j} which connects vertices of this bigon (see Figure 60). What remains of the disks D′_{j} we now call F_{j}, they are faces of the orbifold Ô′.

As the result of this 3-step procedure the graph

⋃_{j=1}^{r} ∂D_{j}

is transformed into the graph K whose embedding into S(G) is well-defined up to isotopy. The orbifold structure associated with K is described as follows. Suppose that an edge e_{ij} := [v_{ij}, v_{ji}] of K is obtained by collapsing the bigon D′_{i} ∩ D′_{j}. Then we take

θ : e_{ij} → α

where 0 < α ≤ π/2 is the angle between the circles ∂D_{i}, ∂D_{j}. The orbifold Ô′ that we have constructed has the underlying set M̄(G), the singular locus is the union of faces F_{j}, 1 ≤ j ≤ r, and the orbifold structure described as above. This orbifold Ô′ is obtained by removing cuspidal neighborhoods of those cusps of the orbifold O which belong to N. To get actual compactification Ô of the orbifold O we also have to chop off cusps which are outside of N.

19.2 Brooks’ Theorem

Recall that for a geometrically finite Kleinian group G ⊂ Isom(ℍ^{3}) we have the Teichmuller space T (G) which consists of conjugacy classes of isomorphic embeddings ρ : G ↪ Isom(ℍ^{3}) that are induced by quasiconformal homeomorphisms. Let d_{T} denote the Teichmuller metric on T (G).

Theorem 19.2 (R. Brooks, [32]) Suppose that G_{0} ⊂ Isom(ℍ^{3}) is a geometrically finite Kleinian group. Then for any ϵ > 0 there exists a type-preserving monomorphism ρ_{ϵ} : G_{0} → G_{ϵ} ⊂ Isom(ℍ^{3}) so that:

• [ρ_{ϵ}] ∈ T (G_{0}), d_{T} (id, ρ_{ϵ}) ≤ ϵ;

• G_{ϵ} ⊂ Γ where Γ is a lattice in Isom(ℍ^{3});

• The convex hull CΛ(G_{ϵ}) is precisely invariant under the subgroup G_{ϵ} of Γ;

• If the orbifold ℍ^{3}/G_{0} is locally-reflective, so is the orbifold ℍ^{3}/Γ. If any element of finite order in G_{0} is a reflection, then ℍ^{3}/Γ is an all right alternated orbifold.

The proof of this theorem occupies the rest of this section.

Suppose that D_{1}, ..., D_{n} is a collection of closed round disks in \overline{ℂ} so that each pair of these disks is either disjoint or tangent. Each complementary region C to D_{1} ∪ ... ∪ D_{n} ⊂ C is called an ideal circular polygon. The vertices of C are points of tangency between the disks which belong to cl(C).

Lemma 19.3 Suppose that C is an ideal circular polygon in \overline{ℂ}. Then there exists a finite collection of disks D_{n+1}, ..., D_{n+N} such that each component of

C − (D_{n+1} ∪ ... ∪ D_{n+N})

is either an ideal triangle or ideal rectangle.

Proof: Assume that the number of ideal vertices of C is at least 5. We start with an ideal vertex v of C formed by the disks D_{1}, D_{2} and inscribe in this corner of C a round disk D which is tangent to D_{1}, D_{2} and is disjoint form all other disks D_{j}. Then start moving D away from v keeping it tangent to D_{1}, D_{2} until D will touch another disk D_{3}. If D_{3} is disjoint from D_{1} and D_{2} then we stop. Otherwise assume that (say) D_{1} is tangent to D_{3} (see Figure 61).

In this case we continue moving the disk D so that D ⊂ C and D is tangent to D_{2} and D_{3}. We stop when D will touch another disk (say) D_{4}. It’s easy to see that the disk D constructed this way separates the domain C into 3 ideal polygons C_{1}, C_{2}, C_{3}. One of them (bounded by boundary arcs of D, D_{1}, D_{2}, D_{3}) is an ideal rectangle. Direct computation shows that the number of ideal vertices of each polygon C_{j} is less than n (see Figure 62).

The same is true if we stop after the disk D touches D_{3} which is disjoint from D_{1}, D_{2}. Let D_{n+1} := D. Thus, arguing by induction, we construct the family of disks D_{n+1}, ..., D_{n+N}. □

Note that we can’t possibly find a collection of round disks so that all complementary regions are ideal triangles. To do this one needs to deform the original polygon C as well.

Consider an ideal circular quadrilateral Q ⊂ \overline{ℂ}. We call this quadrilateral injective if the round circles containing edges of Q intersect only at vertices of Q (see Figure 63). Thus, if H is the group generated by reflections in these four circles, H has fundamental domain consisting of the union of the interior quadrilateral Q and the exterior quadrilateral Q̄, hence it is quasiconformally conjugate to a Fuchsian group (where all four circles are orthogonal to a straight line). We conclude that the group H is quasifuchsian .

We label the sides of Q as: top, right, bottom, left, so that this labelling is consistent with the natural orientation of ∂Q, otherwise the labelling has nothing to do with any particular coordinate system in \overline{ℂ} (see Figure 63).

Note that we can always inscribe into Q a round circle C which is tangent either to the top, bottom and left sides of Q (following [31] we call it a “horizontal” circle despite the fact that it rather looks vertical to me) or to the top, left and right sides (we call it a “vertical” circle). If this circle is horizontal and vertical simultaneously, then we call this circle “neutral”. If C is not neutral then it splits Q into two ideal triangles and one ideal quadrilateral Q′. Three sides of Q′ already have labels coming from the original quadrilateral Q, there is a unique way to label the remaining side so that Q′ gets a topright-bottom-left labelling. If C is a neutral circle then we stop, otherwise we inscribe a horizontal or vertical round circle into Q′, etc.

As the result we get a finite or infinite circle packing of Q. Each point of Q either belongs to a closed disk bounded by one of the circles of this packing or to an ideal triangle bounded by arcs of these circles, or it is a point of accumulation of the circles. Note that since the Euclidean diameter of the sequence of quadrilaterals tends to zero, there is only

Figure 63: Q is the interior and Q̄ is the exterior quadrilaterals .

one point of accumulation of circles. This point could possibly belong to one of the circles in the packing. We denote the resulting packing P := P(Q). Now we associate a number r(Q) ∈ (0, ∞), which is called the packing invariant of Q, as follows.

Let n_{1} be the number of horizontal circles ∈ P that we can inscribe until we will need a vertical circle. If one of these circles is neutral then we count it as a horizontal circle and let r(Q) := n_{1}. Otherwise let n_{2} be the number of vertical circles ∈ P that one can inscribe until one again needs a horizontal one. As before, if one of these circles is neutral, then we count it as vertical and let

r(Q) := n_{1} + \frac{1}{n_{2}}

Otherwise we continue this process. As the result we get a real number 0 < r(Q) < ∞, whose continued fraction expansion is:

r(Q) := n_{1} + \frac{1}{n_{2} + \frac{1}{n_{3}+...}}

Figure 64: Degenerate quadrilaterals .

This is the packing invariant of Q.

Now we generalize this invariant to the cases of degenerate quadrilaterals Q. Namely, if top and bottom sides of Q are tangent then we let r(Q) = ∞, if left and right sides are tangent then we let r(Q) = 0. See Figure 64.

Consider the configuration space C(4) of labelled ideal quadrilaterals in \overline{ℂ}. We assume that circles containing boundary arcs of elements Q ∈ C(4) do not degenerate to points, but we allow degeneration of quadrilaterals as on Figure 64: either top and bottom sides or left and right sides could be tangent. The space C(4) has a natural topology. The packing invariant defines a function

r : C(4) → [0, ∞]

Figure 65: Packing invariant for degenerate quadrilaterals .

Lemma 19.4 The function r is continuous.

Proof: We first consider the cases of degenerate quadrilaterals . Suppose that Q is a degenerate quadrilateral where top and bottom sides are tangent. Then we can inscribe infinitely many horizontal circles C_{j} into the “left” half of the quadrilateral Q. Now we “regenerate” Q to a quadrilateral Q_{ϵ}, where ϵ is the smallest distance between top and bottom sides of Q_{ϵ} (see Figure 65). It follows from elementary calculations that one can perturb ≈ const/ϵ circles C_{j} so that they become horizontal circles inscribed into Q_{ϵ}. Here const is a constant which depends only on the geometry of the original configuration. Thus

lim_{ϵ→0} r(Q_{ϵ}) = ∞ = r(Q)

Similar argument works in the case when left and right sides of Q are tangent.

Now we can consider the general case. Suppose that Q is a nondegenerate quadrilateral with an infinite circle packing P, r := r(Q). Suppose that δ > 0, N = [1/δ]; then we can find a neighborhood U of Q in C(4) such that any Q_{ϵ} ∈ U has the same first N numbers n_{1}, ..., n_{N} describing combinatorics of circle packing of the quadrilateral Q. Thus |r(Q_{ϵ}) − r(Q)| ≤ const/N. We conclude that the function r is continuous in such points Q. The last case we have to consider is when Q has a finite circle packing P(Q). Let C_{k} denote the last (neutral) circle in P, where the numbers n_{1}, ..., n_{k} describe combinatorics of the circle packing of Q. Small perturbations of Q can move Q to either a horizontal or vertical circle (depending on a sequence Q_{ϵ} in C(4) convergent to Q). Let me assume that C_{k} becomes horizontal after this perturbation. Then we get a a sequence of quadrilaterals Q_{k,ϵ} ⊂ Q_{ϵ}, where C_{k} corresponds to the top or bottom side, ϵ → 0 is the distance between top and bottom sides. As ϵ → 0 the quadrilaterals Q_{k,ϵ} degenerate to a quadrilateral Q_{k} where top and bottom sides of Q_{k} are tangent. Thus we are in the situation when we can apply our analisys of continuity of the function r at degenerate quadrilaterals . Namely,

r(Q_{ϵ}) ≈ n_{1} + \frac{1}{n_{2} + _{\frac{1}{n_{3}+... \frac{1}{n_{k}+ ϵ/const}}}} ≈ r(Q)

The case when C_{k} becomes vertical in Q_{k,ϵ} is similar. □

Consider a configuration C of 4 tangent round disks with labels as on Figure 63. They bound two ideal quadrilaterals : Q and Q̄. Both quadrilaterals have some packing invariants r(Q), r(Q̄).

Proposition 19.5 The pair (r(Q), r(Q̄)) ∈ ℝ_{+} × ℝ_{+} uniquely determines the configuration C up to a Moebius transformation γ : \overline{ℂ} → \overline{ℂ}.

Proof: I have to say that I am quite uncomfortable with the proof of this statement presented in [31]. Thus I will reduce this Proposition to a more recent rigidity theorem of Oded Schramm. Consider the circle packing of the complex plane \overline{ℂ} given by P(Q) ∪ P(Q̄). Only two points in \overline{ℂ} are points of accumulation for these packings, thus at most two points in \overline{ℂ} are not in the union of corresponding round disks and ideal triangles. (In the terminology of [175], complement to the carrier of P(Q) ∪ P(Q̄) consists of at most two points.) If we have another configuration C′ with the same pair of packing invariants (r(P(Q′)), r(P(Q̄′))) then the packings P(Q) ∪ P(Q̄) and P(Q′) ∪ P(Q̄′) are combinatorially equivalent and therefore are Moebius-equivalent according to [175]. Thus there is a Moebius transformation which carries C to C′. □

We associate a discrete Moebius group with the packing P = P(Q) of each ideal quadrilateral Q as follows. Let C denote the configuration of four round circles which contain edges of Q. For each ideal triangle T_{j} formed by arcs of circles in P we take an extra circle C(T_{j}) which contains its vertices. As the result we get a pattern of circles D which contains circles of C, P and all C(T_{j}), j = 1, 2, .... Each pair of circles in D is either tangent, orthogonal or disjoint. Then take the group Γ_{Q} generated by reflections in all circles of D. Similarly to Section 19.1, the convex polyhedron Φ ⊂ ℍ^{3} bounded by convex hulls of circles in D is a fundamental domain for the group Γ_{Q}. A fundamental domain for the action of Γ_{Q} on \overline{ℂ} is the quadrilateral Q̄ = int(∂_{∞}Φ).

Remark 19.6 Note that ∂_{∞}Φ − Q̄ consists of a countable number of ideal vertices.

Take two configurations C, C′ of four tangent round circles in \overline{ℂ} as above, Q, Q′ are ideal quadrilaterals that they bound.

Proposition 19.7 Suppose that r(Q) = r(Q′). Then there exists a quasiconformal homeomorphism f : \overline{ℂ} → \overline{ℂ} which carries Q to Q′, P(Q) to P(Q′) and D to D′.

Proof: Our proof again follows [31], Lemma 4.1. However in the final part of the proof I will use the Compactness Theorem 14.93 to complete the argument. Let R_{x} ⊂ ℂ denote the Euclidean rectangle with the vertices {0, x, x + i, i}. This rectangle has the natural top-right-left-bottom labelling of the sides. The labelled quadrilateral Q̄ is nondegenerate, thus there is a Riemann mapping ϕ : Q̄ → R_{a}, which sends vertices to vertices and preserves labels. The number a is the conformal modulus of Q̄. For any other rectangle R_{b} define the affine map

A_{b} : z ↦ Re(z)b/a + iIm(z), A_{b} : R_{a} → R_{b}

Then we get a Beltrami differential μ_{b} on Q̄ given by ϕ^{∗}(\overline{∂}A_{b}/∂A_{b}). Recall that Q̄ is a fundamental domain for the action of Γ_{Q} on \overline{ℂ}. To construct the corresponding quasiconformal map h_{b} : \overline{ℂ} → \overline{ℂ} we disperse the Beltrami differential μ_{b} from Q̄ to Ω(Γ_{Q}) via the action of Γ_{Q} and let it be zero on the limit set. We retain the notation μ_{b} for the resulting Beltrami differential . As in the proof of Bers’ Theorem 9.26, we let h_{b} be a normalized solution of the Beltrami equation

μ_{b}∂h_{b} = \overline{∂}h_{b}

This quasiconformal map will conjugate Γ_{Q} to another discrete Moebius group Γ_{b}. Note that circles in D(Q) are fixed-point sets of involutions in Γ_{Q}. Thus h_{b} carries circles from D := D(Q) to round circles in D_{b} and will preserve combinatorics of the circle packing. Whence, we get a 1-parameter family D_{b} of quasiconformally equivalent circle packings so that the conformal modulus of the quadrilateral Q̄_{b} equals b. The parameter b varies from 0 to ∞. Let ρ_{b} denote the isomorphic embedding of Γ_{Q} into Isom(ℍ^{3}) induced by h_{b}.

Lemma 19.8 Q̄_{b} is convergent to degenerate quadrilaterals in C(4) as b → 0, b → ∞.

Proof: Take the first k circles C_{1}, ..., C_{k} in P(Q) among which there is at least one horizontal and one vertical circles (one of them could be neutral). We will consider the case when none of these circles is neutral, the remaining case is done similarly. The finite partial packing P̂^{k} := C ∪ C_{1} ∪ ... ∪ C_{k} has two complementary quadrilaterals : Q̄ and Q_{k}. Let Γ_{k} be the subgroup of Γ_{Q} generated by reflections in circles of P̂^{k}.

By our choice of the number k, no pair of opposite sides of Q̄ corresponds to circles which give a pair of sides of Q_{k} (see Figure 66).

Thus the hyperbolic orbifold O := \overline{ℍ^{3}/Γ_{k}} is acylindrical and we can apply the Compactness Theorem 14.93 to conclude that the sequence of discrete and faithful representations ρ_{b} is convergent to a representation ρ_{0} if b → 0 and is convergent to ρ_{∞} if b → ∞. Hence we get convergence of Q̄_{b} in C(4) as well. Clearly each of these limits is a degenerate quadrilateral since the conformal modulus depends continuously on Q̄_{b} ∈ C(4). □

Using Lemma 19.4 and Lemma 19.8 we conclude that r(Q̄_{b}) varies continuously as a function of b from 0 to +∞. Hence we can find a value of b such that r(Q̄_{b}) = r(Q̄′). Lemma 19.5 implies that the configurations C_{b} and C′ are Moebius-equivalent. By construction there exists a quasiconformal homeomorphism of \overline{ℂ} which carries D to D_{b}. This proves Proposition 19.7. □

Suppose now that G_{0} is a geometrically finite discrete subgroup of Isom(ℍ^{3}) so that:

Figure 66: Partial packing.

• a fundamental domain for the action of G_{0} in \overline{ℂ} consists of a finite number of ideal injective quadrilaterals Q_{01}, ..., Q_{0m};

• the group G_{0} contains reflections in edges of these quadrilaterals .

Choose a top-right-bottom-left labelling for edges of the quadrilaterals Q_{01}, ..., Q_{0m}. Let [ρ] ∈ T (G_{0}) be a representation whose image is a quasiconformal deformation G of the group G_{0}. Then G again has a fundamental domain Q_{1} ∪ ... ∪ Q_{m} and the quasiconformal conjugation of G_{0} and G gives labels to edges of these quadrilaterals and establishes a bijection Q_{0j} → Q_{j}. Each of the quadrilaterals Q_{j} has the packing invariant r(Q_{j}). Hence we get a function:

F : T (G_{0}) → (ℝ_{+})^{m}, F (ρ) = (r(Q_{1}), ..., r(Q_{m}))

By abusing notations we will write down this function as F (G) when G = ρ(G_{0}) and it is clear what the “marking” ρ : G_{0} → G is.

Theorem 19.9 (R. Brooks) The function F is continuous and injective.

Proof: The quadrilaterals Q_{j} vary continuously with the representation ρ, thus continuity of F follows from Lemma 19.4. The nontrivial part of Theorem is injectivity of F. Suppose that that G, G′ are discrete groups which are quasiconformal deformations of G_{0} so that F (G) = F (G′). By Proposition 19.7 there are quasiconformal homeomorphisms ϕ_{j} : Q_{j} → Q′_{j} which carry packings P(Q_{j}) to P(Q′_{j}) and D to D′. Unions of quadrilaterals Q_{j} (and Q′_{j}) are fundamental domains for G and G′. Using actions of the groups G, G′ we extend the homeomorphisms ϕ_{j} to an equivariant quasiconformal homeomorphism ϕ : \overline{ℂ} → \overline{ℂ}. As before we extend the groups G, G′ by adding reflections in circles of D(Q_{j}), D(Q′_{j}), j = 1, ..., m. The resulting groups Γ, Γ′ are discrete, their limit sets equal to \overline{ℂ}. Recall that the groups G, G′ are geometrically finite and each packing P(Q_{j}), P(Q′_{j}) has only one accumulation point in \overline{ℂ}. Thus the groups Γ, Γ′ have convex fundamental polyhedrons with only countably many ideal vertices. Hence the action of Γ on its limit set is recurrent (see Theorem 9.25). The quasiconformal homeomorphism h conjugates Γ to Γ′ and we apply Sullivan Rigidity Theorem 9.23 to conclude that h is Moebius. Therefore G = G′ define the same point in T (G_{0}). □

Corollary 19.10 The map F is a local homeomorphism.

Proof: Let Q_{j} be one of the ideal injective quadrilaterals in the fundamental domain of G. The group G_{j} generated by reflections in edges of Q_{j} is the stabilizer of a component of Ω(G) which contains Q_{j}. Since Q_{j} is injective, the group G_{j} is quasifuchsian . Hence, according to Section 9.12, the Teichmuller space T (G) is homeomorphic to ℝ^{m}. Thus F is an injective continuous map ℝ^{m} → ℝ^{m} and must be a local homeomorphism. □

Finally we can finish the proof of Theorem 19.2. We have to construct a finite pattern of round disks in S(G_{0}) (in the sense of Section 19.1) so that complementary regions to the union of disks in the pattern are ideal quadrilaterals and triangles. To do it we first take care of possible cusps in the surface S(G_{0}). Consider a round disk D in S(G_{0}) which is a neighborhood of a cup in S(G_{0}). By taking an appropriate conjugation of the group G_{0} in P SL(2, ℂ) we can assume that D is covered by an upper half-plane D̃ = {z : Im(z) > t} ⊂ Ω(G_{0}). The stabilizer of D̃ in G_{0} is a cyclic parabolic group A generated by the horizontal translation α : z ↦ z + 1. The group A has a fundamental domain

Ψ := {z : 0 < Re(z) < 1}

We inscribe into Ψ the round disk Δ̃ := {z : |z − it + 1/2| ≤ 1/2}. See Figure 67. By choosing t sufficiently large we guarantee that the interior of Δ̃ projects homeomorphically into S(G_{0}). Let Δ denote the projection. This way we choose pairs of round disks D_{j}, Δ_{j} for each cusp P_{j} in S(G_{0}) so that disks corresponding to distinct cusps are disjoint.

Figure 67: Filling in cusps.

As the next step we use Lemma 19.3 to construct a collection of round disks in S(G_{0}) − ∪_{j}(D_{j} ∪ Δ_{j}) which have disjoint interiors and so that each complementary region is either an ideal triangle T_{k} or an ideal quadrilateral . Let Q_{01}, ..., Q_{0m} be the complete list of complementary ideal quadrilaterals . For each complementary ideal triangle T_{k} we choose a round disk in S(G_{0}) whose boundary contains the vertices of T_{k}. By taking the union of all round disks which we have constructed, we get a pattern Q of round disks in S(G_{0}). Now we associate with this pattern a discrete geometrically finite group Ĝ_{0} ⊂ Isom(ℍ^{3}) as in Section 19.1. (I recall that we lift all the round circles of the pattern to Ω(G_{0}) and add reflections in these circles to the group G_{0}.) The group Ĝ_{0} satisfies the conditions of Theorem 19.9. We retain the notations Q_{01}, ..., Q_{0m} for lifts of the quadrilaterals from S(G_{0}) to ideal quadrilaterals in the domain Ω(G_{0}). It follows from the description of fundamental polyhedron of reflection group in the Section 19.1, that Q_{01} ∪ ... ∪ Q_{0m} is a fundamental domain for the group Ĝ_{0}. Rational vectors are dense in ℝ^{m}, thus by Corollary 19.10 there is a dense set E of representations [ρ] ∈ T (Ĝ_{0}) such that F (Ĝ = ρ(Ĝ_{0})) are rational vectors. Here F is the function given by packing invariants of the quadrilaterals Q_{0j} as above.

Let [ρ] ∈ E, then P(Q_{j}) are finite circle packings of Q_{j}’s. As in the proof of Theorem 19.9 we associate with each packing P(Q_{j}) a discrete group Γ_{Q_{j}} which is geometrically finite since P(Q_{j}) is finite. The group Γ generated by Γ_{Q_{j}}, 1 ≤ j ≤ m and Ĝ is geometrically finite and has empty domain of discontinuity. Hence this group is a lattice in Isom(ℍ^{3}).

The rest of the assertions of Theorem 19.2 are easy to verify. Note that we add to the group G_{ϵ} reflections in hyperbolic hyperplanes which are disjoint from the convex hull of Λ(G_{ϵ}). Thus a fundamental domain of the action of G_{ϵ} on CΛ(G_{ϵ}) is contained inside a fundamental domain of the lattice Γ, which implies that CΛ(G_{ϵ}) is precisely invariant under G_{ϵ}. By construction, if ℍ^{3}/G_{0} is locally reflective, then ℍ^{3}/Γ is also locally reflective. Suppose that each element of finite order in G_{0} is a reflection. Then fixed-point sets of distinct reflections in G_{0} are disjoint in ℍ^{3}. So we label them “white”. According to construction of the group Γ we first add to the group G_{ϵ} a collection of reflections whose fixed-point sets are disjoint (and they are also disjoint from fixed-point sets of reflections from G_{0}), label them “white” as well. On the last step of the construction we “fill in” ideal triangles in a packing of Ω(G_{ϵ})/G_{ϵ} by round discs. The corresponding generation of hyperbolic hyperplanes consists of disjoint planes which intersect “white” hyperplanes by right angles. So, we label the last generation of hyperplanes “red”. □

19.3 Construction of manifold coverings over alternated orbifolds

Suppose that O is a locally-reflective 3-dimensional orbifold, I will assume that interior of O admits a complete hyperbolic structure and |O| is orientable. The universal cover ℍ^{3} of O has a partition into convex domains which are inverse images of int(|O|) under the projection ℍ^{3} → O. If C is the closure of one of these domains and G := G_{C} is its stabilizer in π_{1}(O), then G_{C} acts freely on C and |O| = C/G_{C} (see Section 7.5). Let R := R_{C} be the subgroup in π_{1}(O) generated by reflections in faces of C. Clearly gRg^{−1} = R for every g ∈ G. According to Poincare’s Theorem on fundamental polyhedra, C is a fundamental domain for the action of R on ℍ^{3}. We conclude that the fundamental group π_{1}(O) is generated by R and G; hence π_{1}(O) = R ⋉ G. In particular g ∈ π_{1}(O) is orientation-reversing iff its a product of an odd number of reflections in R and an element of G.

Suppose that γ ≠ 1 is an element of finite order. Then there is an element g ∈ G such that F ix(gγg^{−1}) ∩ C ≠ ∅. This implies that (up to conjugation) either γ is a reflection or is a rotation of order 2, which is the product of two reflections in R.

Our next goal is to give an explicit construction of a finite index torsion-free subgroup in π_{1}(O) provided that O is an all right alternated orbifold. Recall that this implies that all reflections in R are assigned one of two colors: white and red. The action of G by conjugations preserves this coloring. We define an epimorphism ψ : π_{1}(O) → ℤ/2 × ℤ/2 as follows. The group ℤ/2 × ℤ/2 has two generators of order two, let’s call them “white” and “red”. We send reflections in R to generators of ℤ/2 × ℤ/2 which have the same color. Note that the group R has the presentation

⟨τ_{j} : τ_{j}^{2} = 1, (τ_{i}τ_{j})^{2} = 1, i, j = 1, 2, ...⟩

where the index j runs over all faces of C, τ_{j} is the reflection in that face and we have relations (τ_{i}τ_{j})^{2} = 1 iff the corresponding faces intersect by the right angle (which implies that they have distinct colors). Thus we have defined a homomorphism ψ : R → ℤ/2 × ℤ/2. Extend this homomorphism to the group G as ψ(G) = {1}. Since conjugations by elements of G preserve white/red labels, we have a well-defined epimorphism ψ : π_{1}(O) → ℤ/2× ℤ/2. Because of our description of elements of finite order in π_{1}(O), the kernel K of ψ is torsionfree. This gives us a regular 4-fold manifold covering p : N → O. Clearly N is orientable.

We need to understand this covering over boundary components of

Ō = (ℍ^{3} ∪ Ω(π_{1}(O)))/π_{1}(O).

Recall that each component S_{j} of Ω(π_{1}(O)))/π_{1}(O) is a locally reflective 2-orbifold without corner reflectors so that all components of the singular locus Σ_{S_{j}} inherit the same labels (say white) from the orbifold O. Therefore ψ : π_{1}(S_{j}) → ℤ/2, p^{−1}(S_{j}) consists of two components S′_{j}, S′′_{j}; each component is the double of |S_{j}| along Σ_{S_{j}}. Hence each covering S′_{j} → S_{j}, S′′_{j} → S_{j} is characteristic, i.e. if τ : S_{j} → S_{i} is a diffeomorphism then τ lifts to a diffeomorphism

τ̃ : S′_{j} ∪ S′′_{j} → S′_{i} ∪ S′′_{i}

19.4 Construction of finite type hyperbolic orbifolds

In this section we start the proof of Theorem 15.2, which will be finished in the Section 19.6.

We have a compact orientable atoroidal Haken paired manifold (M′, P′) and a connected superincompressible surface S′ ⊂ M′. I recall that P′ is a collection of incompressible boundary tori and annuli. In each of the annuli we draw the middle circle. The compact manifold M is obtained from M′ by splitting along S′. As the result we get the following combinatorial data on the boundary of M:

• A surface L ⊂ ∂M and an involution τ : L → L so that after gluing along τ we get M/τ = M′, L/τ = S′.

• A collection P of subsurfaces in ∂M − L which we get after splitting P′ along S′; P consists of tori, annuli and squares. The latter have distinguished segments A_{j} connecting opposite sides (these segments correspond to the middle circles on cylindrical components of P′).

Given this data we will construct an alternated orbifold O of finite type with the underlying space M. Gluing of O via τ gives rise to an orbifold O′ of the class R with the underlying space M′.

Figure 68:

Let Q be the collection of toroidal and cylindrical components of P. By the hypothesis of Theorem 15.2, the paired manifold (M, Q) admits a complete geometrically finite hyperbolic structure, where Q is the designated parabolic locus. As usual there are two cases depending on the number of components in M, I will consider the case when M is connected, the other case is absolutely similar. Then there exists a geometrically finite torsion-free Kleinian group G ⊂ P SL(2, ℂ) with the quotient

M̄(G) = [ℍ^{3} ∪ Ω(G)]/G

The manifold M is a Scott compact core of M̄(G). Components of Q correspond to cusps of the manifold M̄(G). We assume that M ∩S(G) = Ω(G)/G is a subsurface S_{c} obtained by removing from S(G) punctured disks which are cuspidal neighborhoods . Then the compact core M is given by Theorem 6.82.

Note that L ⊂ S_{c} is an incompressible surface so that no component of L^{∗} := S_{c} − L is an annulus (which means that if a boundary component of L is isotopic to a boundary curve of Q, then these loops are actually equal). See Figure 68.

Recall that we have square components of P −Q. Each of these components has exactly two opposite boundary segments which belong to L; these segments are connected by arcs A_{j}, we call these arcs parabolic (since in the manifold M′ they correspond to the parabolic locus). We extend this collection of arcs to A := A_{1} ∪ ... ∪ A_{k}, which is a collection of disjoint, properly embedded arcs in the surface (L^{∗}, ∂L^{∗}). Some of these arcs (1 ≤ j ≤ p) are parabolic, the rest are called hyperbolic arcs (it may happen that either the set of parabolic or hyperbolic arcs is empty). We assume that the complement L^{∗} − A consist of simply-connected components.

Figure 69: Blowing up the vertex v.

The union B := A ∪ ∂L^{∗} is a graph on the surface S_{c}. After adding more vertices to B, we assume that the restriction of τ to ∂L^{∗} ∩ ∂L preserves the cellular structure given by this graph. We also assume that no two vertices of valency 3 on B are adjacent. For all parabolic arcs A_{j} we have their regular neighborhoods N (A_{j}) in L^{∗} which are just square components of P − Q.

We form another graph B̂ as follows. For every end-point v of each hyperbolic arc we take the end-points v_{2}, v_{3} ∈ ∂L^{∗} and v_{1} of the edges emanating from v and connect them to form a triangle as on Figure 69. Then we erase the vertex v and the edge [v_{1}, v] from the graph. As the result we “blow up” the vertex v into a hyperbolic triangle [v_{1}, v_{2}, v_{3}].

Figure 70: Collapse of a regular neighborhood of a parabolic arc A_{j} to a parabolic segment I_{j}.

At the next step we collapse regular neighborhood N (A_{j}) of each parabolic arc A_{j} along the family of segments parallel to A_{j}. See Figure 70. As the result all what is left from the rectangle N (A_{j}) is a parabolic segment I_{j}, we erase all interior vertices of these segments. The result is the graph B̂ ⊂ S_{c}.

Suppose that P = {B_{j}, j = 1, ..., r} is a partial packing of the surface S(G) (i.e. it is a collection of round disks in in S(G) whose interiors are pairwise disjoint). The combinatorial structure of P is described by its dual graph P^{#} ⊂ S(G) which is defined as follows. Vertices of P^{#} are points in the interior of B_{j} (one vertex for each disk). Edges of P^{#} are segments connecting distinct vertices such that corresponding closed disks have nonempty intersection. Note that embedding of this graph into S(G) is well-defined up to isotopy.

Theorem 19.11 For the given Kleinian group G, surfaces L, S_{c}, graph B, etc., as above, there exists a subdivision B′ of B and a partial packing P of S_{c} such that:

• The graph B̂′ is isotopic to a graph P^{#} ⊂ S_{c} which is dual to P = {B_{j}, j = 1, ..., r};

• After isotopy the involution τ : L → L preserves the cellular structure of B̂′ ∩ ∂L.

Proof: After subdividing B, we can assume that the graph B̂ is a cell complex where cells are points and circular segments in S. After applying an isotopy in S_{c} wecan assume that each hyperbolic triangle and parabolic segment is contained in a small round disk in S_{c}. Then we put round disks with centers at the vertices of parabolic segments and hyperbolic triangles of B̂ so that these disks are either disjoint or tangent (the latter happens if vertices belong to the same parabolic segment or hyperbolic triangle). Next we put small disjoint round disks in each valency 2 vertex of B. We assume that these disks are so small that they are disjoint from any previously constructed disks. In the isotopy class of τ we choose an involution which preserves the cellular structure of B̂, carries B̂∩ L onto itself and sends vertices of hyperbolic triangles to vertices of hyperbolic triangles and parabolic edges to parabolic edges.

Then, we subdivide the portion of the graph B̂ which is not already covered by disks and cover these edges by round disks, each of which are either disjoint or tangent. By choosing sufficiently fine subdivision we assure that τ preserves the cellular structure of B̂. See Figure 71. □

Note that the surface

S_{c} − ⋃_{B_{j}∈P} B_{j}

consists of a subsurface L^{#} isotopic to L and a collection of contractible components corresponding to L^{∗} − A. Henceforth we let L := L^{#}, thus the cellular structure on ∂L ∩ ∂L^{∗} is given by circular arcs and ideal vertices (points of tangency between round disks in P). After appropriate isotopy we assume that τ : L → L preserves this cellular structure and we retain the notation: L^{∗} := S_{c} − L.

As in Section 19.2, we associate a geometrically finite extension Ĝ ⊂ Isom(ℍ^{3}) of the group G via the partial packing P. Then we apply Brooks’ Theorem 19.2 to the group Ĝ. As the result we find a small deformation G_{ϵ} ⊂ Isom(ℍ^{3}) of the group G so that the corresponding partial round disk packing P_{ϵ} of S(G_{ϵ}) can be extended to a finite pattern of round disks D_{ϵ} on S(G_{ϵ}) which has the properties:

• The union of disks in D_{ϵ} is the surface L^{∗};

• any two disks in this pattern are either disjoint, equal, tangent or their boundaries are orthogonal.

Figure 72:

Theorem 19.12 The compact locally reflective orbifold O associated with D_{ϵ} as in Section 19.1 is alternated and has finite type.

Proof: It is easy to see that it satisfies the Axioms 1, 2 and 3. The orbifold O is very good since it is obtained by chopping off cusps of a geometrically finite hyperbolic orbifold.

What requires verification are the Axioms 4, 5. We will verify the Axiom 5, proof of the Axiom 4 is similar. Consider an arc λ ⊂ L^{∗} with the end-points on the boundary of L^{∗}. Then, unless λ goes around a “black box” (a point of intersection of 3 circles in the pattern which will be blown up into a 3-edged square orbifold), it will intersect at least four bigons formed by the pattern of round disks. These bigons will become edges of the graph Γ associated with the pattern of disks, so we get at least 4 vertices on λ. See Figure 72. □

19.5 Gluing orbifolds of the class R from orbifolds of finite type

Our next step is to glue the orbifold O constructed in the previous section into an orbifold of the class R. Recall that L = ∂O − Q has a natural orbifold structure, where Q is the parabolic locus of O; τ : L → L is a homeomorphism preserving this structure. Let O′ be the orbifold obtained from O by gluing via τ. We let L′ = S′ be the image of L in M′.

We describe the combinatorial structure of the orbifold O′ as follows. The underlying space of O′ is the manifold M′. The image of the parabolic locus Q in M′ is the parabolic locus Q′ of the orbifold O′. Let Γ̂ be the graph of the orbifold structure of O as in section 7.3.

Glue the graph (Γ̂ − L) ⊂ ∂M to itself via τ. The result is a graph Γ̂′ on the boundary of M′. Each vertex of Γ′ has valency 3 and belongs to the boundary of Q′.

The function θ descends to a function

θ′ : edges of (Γ′ := Γ̂′ − Q′) → {π/2}

Figure 73: Possible positions for the loop ∂D.

It gives the space M′ the structure of a locally reflective orbifold O′. The boundary of O′ equals Q′ and consist of Euclidean 2-orbifolds.

Proposition 19.13 The orbifold O′ above is Waldhausen, acylindrical and atoroidal.

Proof: The surface L′ has a natural orbifold structure O_{L′}. The fact that the suborbifold O_{L′} ⊂ O′ is incompressible directly follows from the assumption that L′ is incompressible in the manifold M′. Since O − F(P^{(1)}) is acylindrical we conclude that O_{L′} is also superincompressible. Recall that the manifold M′ = |O′| was irreducible and atoroidal by the assumption. Thus, to verify that O′ is irreducible and acylindrical we need to consider only bad, spherical and Euclidean suborbifolds whose underlying sets are either disks or annuli, or Moebius bands properly embedded in M′.

Suppose that (D, ∂D) ⊂ (M′, ∂M′ − Q′) is a properly embedded 2-disk such that D inherits from ∂M′ an orbifold structure of an elliptic (or Euclidean) 2-orbifold O_{D} which doesn’t bound a compact elliptic (or Euclidean) 3-suborbifold in O′. By altering the orbifold O_{D} within its isotopy class we first minimize the number of points of intersection of ∂D with ∂L′.

The boundary of D is transversal to the graph Γ′, the intersection of D with Γ′ determines the set of vertices on ∂D. The number of these vertices is at most 4 (otherwise the orbifold D is hyperbolic). The boundary ∂D corresponds to a collection of arcs I_{j} on the surface Σ = Σ_{O}. Consider one of these arcs I := I_{j}. Since the number of points of intersection ∂D ∩ ∂L′ is minimal within the isotopy class of O_{D}, we conclude that the arc I must have interior vertices (i.e. points of I ∩ Γ). The number of the interior vertices is at least 2 (by the Axioms 4 and 5 of orbifolds of finite type). Since O_{D} is not hyperbolic, we conclude that the number of the interior vertices of each arc I_{j} is exactly 2 and there are precisely two of these arcs I_{1}, I_{2}. The Axiom 5 implies that both arcs are isotopic into S^{0} relatively to ∂I_{j}. By the above analisys of combinatorial possibilities for the arcs I_{1}, I_{2}, the arc I_{1} satisfies the possibility (i) (on the page 97) iff the second arc I_{2} satisfies the same possibility (i). Hence there are exactly two combinatorial possibilities for the loop ∂D (see Figure 73).

In the 1-st case the loop ∂D is a circuit which goes around one of the mirror squares P_{j} ⊂ Q′ ∩ ∂L′ and it intersects exactly four edges of Γ′: those which emanate from the square P_{j}. Thus D is “parallel” to the suborbifold P_{j} in the parabolic locus of O′. In the second case the orbifold O_{D} bounds a Euclidean suborbifold W in O′, where W is the “solid wedge”. Hence O_{D} is compressible. □

Corollary 19.14 The orbifold O′ belongs to the class R.

Proof: It follows from Theorem 7.14 that O′ is very good. In the following section we will give another proof of the fact that O′ is very good by constructing a finite manifold covering over O′. □

19.6 Hyperbolization of locally reflective orbifolds

As in previous section we assume that O is a locally reflective orbifold of finite type, Q is the parabolic locus of this orbifold, L := ∂O − Q; τ : L → L is a gluing involution involution. In the previous Section we have constructed a geometrically finite hyperbolic structure on (O, P); we also proved that the orbifold O′ = O/τ belongs to the class R, the image Q′ of Q in O′ is the designated parabolic locus of O′. Recall that ∂O′ = Q′.

Theorem 19.15 The orbifold O′ −Q′ admits a hyperbolic structure of finite volume, where Q′ corresponds is the parabolic locus.

Proof: Let N → O denote the finite regular manifold cover of O constructed in Section 19.3, the manifold N is orientable. Let F denote the finite group of covering transformations for N → O, Q̂ be the lift of Q to N. We get a paired manifold (N, Q̂). Note that the relative boundary ∂_{0}N = ∂N − Q has exactly four components. The homeomorphism τ lifts to a homeomorphism τ̂ : ∂_{0}N → ∂_{0}N (see Section 19.3).

Take the manifold N′ obtained by gluing of N via τ̂. The gluing homeomorphism τ̂ is F -equivariant. Thus we have a well-defined action of F on N′ so that N′/F = O′. The manifold N′ is Haken, atoroidal and Q̂′ := ∂N′ consists of incompressible tori (lifts of the Euclidean orbifolds of the boundary Q′ = ∂O′).

Therefore the gluing τ̂ : N → N resulting in the manifold N′, satisfies the assumptions of the Theorem 15.1 (Case A). The paired manifold (N, Q̂) clearly is not an interval bundle over surface; note that the image of ∂_{0}N in N′ consists of two components, that is why we have to treat the case of a disconnected superincompressible surface in Theorem 15.1.

Hence, according to Theorem 15.1 (Case A), the paired manifold (N′, Q̂′) admits a complete hyperbolic metric of finite volume. By Mostow Rigidity Theorem 2.46, the action of the finite group F on N′ is homotopic to an action of a finite group of isometries Φ of the hyperbolic metric. Hence the orbifolds O′ and N′/Φ have isomorphic fundamental groups and the orbifold (N − Q̂′)/Φ is complete hyperbolic (see the proof of Theorem 9.22). Namely, if N′ − Q̂′ = ℍ^{3}/G, where G is a torsion-free lattice in P SL(2, ℂ), then we lift Φ to ℍ^{3} and get a finite extension G^{+} ⊂ Isom(ℍ^{3}) of the group G so that ℍ^{3}/G^{+} = (N′ − Q̂′)/Φ.

Thus Theorem 7.29 implies that the orbifolds N/Φ and O′ are homeomorphic, hence O′ − Q′ admits a complete hyperbolic structure. □

Now we can finish the proof of Theorem 15.2. Theorem 19.15 implies that the paired orbifold (O′, Q′) admits a complete hyperbolic structure of finite volume. Therefore the underlying space X of the orbifold O′ − Q′ admits a metrically complete convex hyperbolic structure of finite volume. However X is homeomorphic to the manifold M′ −P′ −E (where P′ is the parabolic locus of M′ and E is a finite collection of points which correspond to square components of Q′). Since X has finite volume, its Nielsen’s convex core C(X) also has finite volume and finitely-generated fundamental group. Annular and toroidal components of P′ correspond to cusps of the complete hyperbolic manifold C(X). Thus we have constructed a geometrically finite hyperbolic structure on the paired manifold (M′, P′). □

19.7 Hyperbolic design

Theorem 19.16 Suppose that G is a finitely-generated discrete subgroup of P SL(2, ℂ). Then there is a geometrically finite group Γ ⊂ P SL(2, ℂ) and a type-preserving isomorphism ρ : Γ → G.

Remark 19.17 The same theorem is valid for discrete finitely-generated subgroups of Isom(ℍ^{3}).

Proof: Let G^{0} be a normal torsion-free subgroup of finite index in G; F := G/G^{0}. According to Theorem 6.85, there exists a relative compact core M^{c} ⊂ M(G^{0}) which is invariant under the action of F. Thus we get an orbifold O = O_{G} = M^{c}/F whose fundamental group is isomorphic to G. Cusps of ℍ^{3}/G define the parabolic locus P ⊂ ∂O_{G}. Note that (O, P) doesn’t have to be a paired orbifold, for instance it could be that O contains a properly embedded essential ANNULUS one of whose boundary components is in P and the other is not. Nevertheless, we can assume that there are no essential ANNULI in O whose boundary components are in P. Note that if ∂O = P, then ∂M^{c} consist only of tori, which means that G^{0} is a lattice and there is nothing to prove. So we shall assume that ∂O ≠ P.

Step 1. First we reduce the problem to the class of discrete groups G such that ∂O−P is incompressible. Suppose that we have proved Theorem for all orbifolds with incompressible boundary. Then the proof is finished by induction on the number of nonparallel compressing DISKS for ∂O − P. Suppose that (O_{1}, P_{1}), (O_{2}, P_{2}) are orbifolds with designated parabolic loci P_{1}, P_{2}, and Γ_{1}, Γ_{2} are geometrically finite Kleinian groups such that the isomorphisms Γ_{j} → π_{1}(O_{j}) preserve type of elements. Let D_{1} ⊂ (∂O_{1} − P_{1}), D_{2} ⊂ (∂O_{2} − P_{2}) be DISKS and the orbifold (O, P) is obtained by gluing these orbifolds via a homeomorphism h : D_{1} → D_{2}. Let me first explain what to do in the case when D_{1} ≅ D_{2} are nonsingular disks. Then π_{1}(O) = π_{1}(O_{1}) ∗ π_{1}(O_{2}) = Γ_{1} ∗ Γ_{2}. Thus we take Γ = Γ_{1} ∗ Γ_{2} be a Klein combination of the groups Γ_{1}, Γ_{2}, the group Γ has the property: the isomorphism Γ → π_{1}(O) preserves type of elements.

Now consider the case when D_{1} ≅ D_{2} is the quotient of the round disk D by the cyclic group J or order m. Then the groups Γ_{j} contain cyclic elliptic subgroups J_{j} which correspond to π_{1}(D_{j}) under the isomorphism π_{1}(O_{j}) → Γ_{j}, j = 1, 2. Choose an isomorphism ϕ : J_{1} → J_{2} of the cyclic groups so that a generator γ_{1} ∈ J_{1} is carried by ϕ to a generator γ_{2} ∈ J_{2} which has the same angle of rotation as γ_{1}.

The elliptic elements γ_{j} have the property: their axes A_{j} contain at least one end-point a_{j} in Ω(Γ_{j}). Take fundamental domains Φ_{j} ⊂ \overline{ℂ} of the groups Γ_{j} so that a_{j} ∈ Φ_{j}. By conjugating Γ_{1}, Γ_{2} in P SL(2, ℂ) we arrange that γ_{1} = γ_{2} and a_{1} ≠ a_{2} are distinct endpoints of the axis A = A_{1} = A_{2}. Next, we take hyperbolic translations h_{1}, h_{2} along A so that h_{j} has the repulsive fixed-point a_{j} and the sets \overline{ℂ}−h_{1}(Φ_{1}), \overline{ℂ}−h_{2}(Φ_{2}) are disjoint. Let Γ_{j} := h_{j}Γ_{j}h_{j}^{−1}, they have fundamental domains Φ_{j} := h_{j}(Φ_{j}) and share a common cyclic elliptic subgroup J = J_{1} = J_{2}. This is the situation when we can apply Maskit Combination Theorem 6.71 and conclude that the group Γ := ⟨Γ_{1}, Γ_{2}⟩ is geometrically finite, isomorphic to Γ_{1} ∗_{J} Γ_{2} and any parabolic element of Γ is conjugate either to Γ_{1} or to Γ_{2}.

This finishes reduction to the case of orbifolds with incompressible boundary.

Step 2. So, we assume that (O, P) has incompressible boundary. Let M := M^{c} and let Q denote the lift of P to ∂M^{c}. Then the manifold M −Q also has incompressible boundary.

Theorem 19.18 There is a geometrically finite Kleinian group Γ^{0} ⊂ P SL(2, ℂ) and an isomorphism ψ : Γ^{0} → π_{1}(M) which preserves type of elements.

Proof: Note that the manifold M is atoroidal (since any ℤ ⊕ ℤ subgroup of G is parabolic). Suppose for a moment that (M, Q) is a paired acylindrical manifold. Take the double DM of this manifold along the partial boundary ∂_{0}M := ∂M − int(Q). The manifold DM is irreducible (since M is irreducible and ∂_{0}M is incompressible), it is Haken (the incompressible surface is ∂_{0}M) and atoroidal (since (M, Q) is acylindrical). Thus we can apply Thurston’s Hyperbolization Theorem to DM and conclude that DM admits a complete hyperbolic metric of finite volume. The paired manifold (M, Q) is not an interval bundle over a surface. Therefore Theorem 13.19 implies that Γ^{0} = π_{1}(M) ⊂ π_{1}(DM) is a geometrically finite subgroup. An element of Γ^{0} is parabolic if and only if it is conjugate to π_{1}(Q). This finishes the proof in the case of acylindrical paired manifolds.

This argument obviously doesn’t work if (M, Q) is either not paired or not acylindrical. There are several ways to deal with this problem. In any case all what we need is a compact (not necessarily connected) paired manifold (or locally reflective orbifold) (B, T) which is irreducible, acylindrical and has incompressible boundary, so that ∂_{0}B = ∂B − T is diffeomorphic to ∂_{0}M = ∂M − Q.

Then we glue the manifold M to B along the boundary ∂_{0}M ≅ ∂_{0}B. The resulting manifold (or locally reflective orbifold) H has the following properties:

• H contains the incompressible surface ∂_{0}M;

• H is irreducible, atoroidal and each boundary component of H is a TORUS.

If H is a manifold, then according to the Hyperbolization Theorem, its interior admits a complete hyperbolic structure of finite volume. The fundamental group π_{1}(M) injects into π_{1}(H) = Γ^{#}. The image under this homomorphism of the fundamental group of each component of Q is a parabolic subgroup of Γ^{#} ⊂ P SL(2, ℂ). Finally, Theorem 13.19 implies that Γ^{0} ⊂ Γ^{#} is a geometrically finite subgroup.

In the case when H is a locally reflective orbifold the situation is slightly more delicate. According to Theorem 7.14, the orbifold H is very good. Thus it admits a finite regular manifold covering H′. Then we apply the Hyperbolization Theorem to H′ and using Mostow rigidity conclude that at least π_{1}(H) is isomorphic to a lattice in Isom(ℍ^{3}) (in fact H itself admits a complete hyperbolic structure). Then we repeat our arguments in the manifold case.

So, what is left is to construct the manifold (or orbifold) B.

1-st way: construction of an orbifold B. We will use Brooks’ Theorem. Namely, let S_{1}, ..., S_{q} be the collection of (closed) boundary surfaces of the manifold M (at this moment we ignore the orbifold locus Q) which have negative Euler characteristic. For each of these surfaces we find a Fuchsian group F_{j} so that ℍ^{2}/F_{j} = S_{j}. Take the inversion σ_{j} in the limit set of F_{j} and consider the group F̂_{j} = F_{j} × ⟨σ_{j}⟩. Clearly Ω(F̂_{j})/F̂_{j} = S_{j}. By Brooks’ Theorem, there exists a small deformation F_{j}^{ϵ} × ⟨σ_{j}⟩ = F̂_{j}^{ϵ} ⊂ Isom(ℍ^{3}) of the group F̂_{j}, so that F̂_{j}^{ϵ} ⊂ Γ_{j}, where Γ_{j} is a lattice in Isom(ℍ^{3}). The convex hull CΛ(F̂_{j}^{ϵ}) is a hyperbolic hyperplane ℍ^{2} ≅ L_{j} ⊂ ℍ^{3}, according to Brooks’ theorem it is precisely-invariant under F̂_{j}^{ϵ} in Γ_{j}. Hence, L_{j}/F̂_{j}^{ϵ} is a boundary component of the underlying set |ℍ^{3}/Γ_{j}|. So we let B_{j} be the orbifold ℍ^{3}/Γ_{j} with “forgotten” orbifold structure on L_{j}/F̂_{j}^{ϵ}. The orbifold B_{j} is a locally reflective orbifold. Boundary of B_{j} consists of the incompressible surface S_{j} and a collection of TORI T_{j}. There is a subgroup G_{j} ⊂ Γ_{j} so that B_{j} − T_{j} = CΛ(G_{j})/G_{j}; boundary of the convex hull CΛ(G_{j}) is the G_{j}-orbit of the hyperbolic plane L_{j}. Clearly B_{j} is atoroidal and irreducible (since it is hyperbolic). Let’s verify that B_{j} is acylindrical. It is enough to check the following:

Suppose that g ∈ G_{j} is such that gL_{j} ≠ L_{j}, then

F_{j} ∩ gF_{j}g^{−1} = {1}  (48)

Suppose that the intersection (48) contains an element γ ≠ 1. Then γ is a hyperbolic element of F_{j}. Therefore the intersection

g(∂_{∞}L_{j}) ∩ ∂_{∞}L_{j}

contains at least two points which are fixed points of γ. However ∂_{∞}L_{j} is a round circle, so gL_{j} ∩ L_{j} ≠ ∅, which contradicts the fact that L_{j} is precisely invariant under F_{j}. Hence B_{j} is acylindrical. Fix a diffeomorphism f_{j} : S_{j} → ∂_{0}B_{j}. The boundary surface S_{j} of M could contain a portion Q_{j} of the designated parabolic locus Q. Let T′_{j} := f_{j}(Q_{j}). The paired orbifold (B_{j}, T_{j} ∪ T′_{j}) is still acylindrical and has incompressible relative boundary ∂_{0}B_{j}. This means that we can take

(B, T) := ⋃_{j=1}^{q}(B_{j}, T_{j} ∪ T′_{j})

This finishes the first construction.

2-nd way: construction of a manifold B.

Theorem 19.19 (R. Myers, [156]) Suppose that M is a compact 3-manifold (possibly with boundary), no boundary component of M has positive Euler characteristic. Then there exists a link L ⊂ int(M) so that the manifold M^{#} := M − Nbd(L) is Haken, atoroidal, acylindrical and has incompressible boundary.

We apply this theorem to the manifold M at hand and let B := M^{#}. □

Step 3. Now we have a geometrically finite Kleinian group Γ^{0} ⊂ P SL(2, ℂ) which is isomorphic to G^{0}. Since ∂_{0}M is incompressible, all components of Ω(Γ^{0}) are simplyconnected. Consider the Teichmuller space

T (Γ^{0}) = T (Σ_{1}) × ... × T (Σ_{m})

where Σ_{j} = Ω_{j}/Γ_{j}^{0}, Ω_{j} are components of Ω(Γ^{0}) stabilized by subgroups Γ_{j}^{0} of Γ^{0}. Note that these subgroups do not have to be quasifuchsian because of accidental parabolic elements. Not all surfaces Σ_{j} are diffeomorphic, let P_{m} ⊂ S_{m} be the subgroup of the group of permutations on m symbols so that each element σ of P_{m} can be realized as a diffeomorphism of M permuting the components of ∂_{0}M according to σ. The finite group F acts on T (Γ^{0}), a fixed point for such action corresponds to a representation ρ : Γ^{0} → Γ^{1} ⊂ P SL(2, ℂ) so that:

• [ρ] ∈ T (Γ^{0});

• Γ^{1} admits a finite extension Γ in P SL(2, ℂ), which is isomorphic to G;

• the isomorphism Γ → G is type-preserving;

(cf. our proof of Theorem 9.22). To prove that F has a fixed point we give T (Γ^{0}) a Riemannian metric d_{W P} which is the direct product of Weil-Petersson metrics on the Teichmuller spaces T (Σ_{j}). The group F acts on T (Γ^{0}) as a subgroup of P_{m} ⋉ (Mod_{Σ_{1}} × ... × Mod_{Σ_{m}}). It follows that F is a finite group of isometries of (T (Γ^{0}), d_{W P}), hence F has a fixed point on T (Γ^{0}) according to Corollary 5.10. □

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