LAMINATIONS, TREES, AND IRREDUCIBLE

AUTOMORPHISMS OF FREE GROUPS

Mladen Bestvina, Mark Feighn, and Michael Handel

Abstract. We examine the action of Out(F_{n}) on the set of (irreducible) laminations. Consequences include a special case of the Tits alternative for Out(F_{n}), the discreteness of certain naturally arising group actions on trees, and word hyperbolicity of certain semidirect products.

§0. Introduction and Preliminaries

The outer automorphism group Out(F_{n}) of the free group F_{n} of rank n naturally maps onto GL_{n}(ℤ) and contains as a subgroup the mapping class group of a compact surface with fundamental group F_{n}. It should not be surprising then to expect Out(F_{n}) to exhibit the phenomena present in both linear groups and mapping class groups. Much of the recent research of Out(F_{n}) has focused on developing tools and proving results known in the other two categories. Culler and Vogtmann have constructed in [CV] a contractible complex “Outer Space” on which Out(F_{n}) acts discretely and with finite stabilizers. This space plays the role of the symmetric space and of the Teichmüller space. As a consequence, Culler and Vogtmann conclude that Out(F_{n}) has type V F L, i.e. a subgroup G of finite index has a K(G, 1) with finitely many cells, thus generalizing the corresponding results for GL_{n}(ℤ) and mapping class groups. In [CM] Culler and Morgan construct an equivariant compactification of Outer Space, much like Thurston’s compactification of Teichmüller space. A point in compactified Outer Space represents an action of F_{n} on a real tree and these in turn play the role of measured geodesic laminations on surfaces.

In [BH] “train-track representatives” of outer automorphisms were constructed. These are analogs of the Jordan normal form and the Nielsen-Thurston normal form. They were used to prove that the fixed subgroup of an automorphism of F_{n} has rank ≤ n.

Hatcher [H] has established another property of Out(F_{n}) already known for GL_{n}(ℤ) and mapping class groups: homological stability. More precisely, if i > 0 is fixed, then for n >> i large the natural homomorphisms Aut(F_{n}) → Out(F_{n}) and Aut(F_{n}) ↪ Aut(F_{n+1}) induce isomorphisms in i^{th} homology.

In this paper we examine the class of irreducible elements of Out(F_{n}). Examples of irreducible outer automorphisms are induced by pseudo-Anosov automorphisms of compact surfaces with only one orbit of boundary component. Generic irreducible automorphisms do not arise in this way, however they should be viewed

Typeset by AMS-TEX

This work was partially supported by NSF grants.

as the natural generalization to Out(F_{n}) of these surface automorphisms. A review of terminology appears later in this section. The mapping class group of a compact surface acts on the space of measured laminations on the surface. In an analogous manner, laminations are defined for irreducible infinite order elements of Out(F_{n}). These laminations arise naturally as limits under forward iteration of a generic conjugacy class in F_{n}. The action of Out(F_{n}) on the (countable) set of such laminations is our focus. We show that stabilizers of points are infinite cyclic, and detect the usual hyperbolic dynamics.

In a series of three papers [BFH1-3], we established a “Tits Alternative” for Out(F_{n}), i.e. subgroups of Out(F_{n}) either contain F_{2} or are virtually finitely generated free abelian. The proof is outlined in an appendix to this paper. The presence of an irreducible element allows for a stronger result whose proof is in Section 4.

Theorem 4.1. Suppose H is a is a subgroup of Out(F_{n}) that contains an irreducible outer automorphism ϕ of infinite order. Then either H contains F_{2} or H is virtually cyclic.

The techniques of this paper together with the combination theorem of [BF2] are used to demonstrate the word hyperbolicity of semidirect products F_{n} ⋊ ℤ induced by infinite order irreducible elements of Out(F_{n}). This is the content of Theorem 5.1. With more care, this may be generalized to show that certain semidirect products F_{n} ⋊ F_{2} are word hyperbolic. See Theorem 5.2. Lee Mosher [Mo] has proved that there are word hyperbolic groups of the form F ⋊ F_{2} where F is the fundamental group of a closed surface.

Associated to an irreducible ϕ ∈ Out(F_{n}) is a ϕ-fixed real tree T_{ϕ}^{+} that is obtained as the limit of the sequence T_{0}ϕ^{i} where T_{0} is any free simplicial tree. Such trees are called irreducible trees. The space of irreducible laminations is then compared to the space of irreducible trees. In the surface theory, there is a perfect correspondence between measured geodesic laminations on a surface and (small) actions of the fundamental group on ℝ-trees [Sk1]. One doesn’t expect the same kind of correspondence in the case of Out(F_{n}) (indeed, the space of (very small) tree actions is finite-dimensional [BF1] [GL], while the natural space of (measured) laminations is infinite-dimensional when n > 2 [Ma]). However, if we restrict our consideration to the irreducible case, then we obtain this perfect correspondence in Corollary 3.6.

Let ϕ be an irreducible element in Out(F_{n}), let T^{+} denote T_{ϕ}^{+} and let T^{−} denote T_{ϕ^{−1}}^{+}. Our final results are the discreteness of certain actions of subgroups of F_{n} on T^{+} × T^{−} and T^{+}. Theorem 5.3 has also been obtained by M. Lustig [Lu2].

Theorem 5.3. The diagonal action of F_{n} on T^{+} × T^{−} is discrete.

Theorem 5.4. Let A be a finitely generated subgroup of F_{n} of infinite index. Then the action of A on T^{+} is discrete.

The techniques used here are those of [BFH1], but the additional hypothesis of irreducibility allows for some strengthening. The most important technical result is Proposition 2.4. Roughly speaking, it states that the lamination associated to an irreducible automorphism “fills” the underlying free group, thus generalizing the familiar fact from the surface theory. This fact alone goes a long way toward establishing that stabilizers of laminations are virtually cyclic (see Proposition 2.6).

The reader interested in the Tits Alternative for Out(F_{n}) [BFH1-3] may find the present paper to be a useful introduction to the techniques of [BFH1]. Indeed, with the irreducibility hypothesis comes a great deal of streamlining. We end this section by reviewing terminology and some facts needed in the sequel.

Irreducible automorphisms.

An (outer) automorphism is reducible if it can be represented as a homotopy equivalence f : G → G on a finite graph (=1-dimensional CW-complex) G without valence 1 vertices that contains an f-invariant proper subgraph G_{0} such that not all components of G_{0} are contractible. Otherwise, the automorphism is irreducible. There exist irreducible automorphisms of finite order, but we will be concerned only with the ones of infinite order. Thus, when we say that an automorphism ϕ is irreducible, we tacitly assume it has infinite order.

Train-track maps.

Let f : G → G be a map on a finite graph G that sends vertices to vertices and is linear on edges. We say that f is a train-track map if there is a collection L of unordered pairs of distinct germs of edges of G satisfying

• both germs in a pair in L emanate from the same vertex of G,

• L is closed under the natural action of f, and

• for every edge e in G, all pairs of germs crossed by the edge-path f(e) are in L.

The elements of L are called legal turns. Pairs of germs incident to the same vertex and not in L are illegal turns. An edge-path in G is legal if it crosses only legal turns.

Every irreducible automorphism has a train-track representative f : G → G. One can assign positive numbers, called lengths, to the edges of G so that f uniformly expands the length of each legal path by a constant factor λ > 1. This number is the Perron-Frobenius eigenvalue of the transition matrix associated with f, the lengths of edges represent the positive eigenvector, and they are unique up to scale.

Nielsen paths, geometric and nongeometric automorphisms.

Let f : G → G be a train-track map representing ϕ ∈ Out(F_{n}). A path in G is Nielsen if it is f-periodic up to homotopy rel endpoints. Every Nielsen path (pulled tight) is a concatenation of indivisible Nielsen paths or iNp’s. An iNp has exactly one illegal turn. A train-track map has only a finite number of iNp’s (and it is possible to arrange the train-track map to have at most one orbit of iNp’s). If they exist, they might close up to yield periodic loops. In that case we say that ϕ is geometric as it can be represented as a (pseudoAnosov) homeomorphism of a punctured surface. If nontrivial periodic loops don’t exist, we say that ϕ is nongeometric. In that case there is an integer m such that it is impossible to concatenate more than m indivisible Nielsen paths.

Proofs of these facts when the period is 1 can be found in [BH, section 4]; the general case is a straightforward modification.

Notation.

If α is a map of a line, a segment, or a loop into a graph, denote by [α] the tightening of α (rel infinity or endpoints, as appropriate). If [αβ] = [α][β] then we write α · β (for finite paths), and call it the concatenation of α and β.

Bounded Cancellation.

Let f : G → G be a train-track map representing ϕ ∈ Out(F_{n}). If α · β is a concatenation of legal paths α and β, but the turn at the dot is illegal, there might be some cancellation between f(α) and f(β). However, the amount is bounded by a constant BCC(f) depending on f (and the chosen metric) but not on α and β. In particular, if α, β, γ are legal and l = length(β) satisfies λl − 2BCC(f) > l, then iteration and tightening of α · β · γ will produce paths with the length of the legal leaf segment corresponding to β going to infinity. This is equivalent to requiring l > \frac{2BCC(f)}{λ−1}, and the number \frac{2BCC(f)}{λ−1} is the critical constant for f. For every C that exceeds the critical constant there is μ > 0 such that if β above has length ≥ C then the length of the legal leaf segment of [f^{k}(α · β · γ)] corresponding to β is ≥ μλ^{k}length(β).

The proof of the existence of BCC(f) can be found in [Co]. A more general statement is contained in Section 3 of the paper at hand.

Relative Train-Track Maps.

Every (reducible) outer automorphism can be represented as a homotopy equivalence f : G → G taking vertices to vertices and edges to edge-paths such that G has a filtration ∅ ⊂ G_{0} ⊂ G_{1} ⊂ ⋯ ⊂ G_{k} = G by f-invariant subgraphs and the following holds for every i. For every edge-path α in \overline{G_{i} \ G_{i−1}} the edge-path [f(α)] is a concatenation of nondegenerate edge-paths in G_{i−1} and \overline{G_{i} \ G_{i−1}} with the first and the last belonging to \overline{G_{i} \ G_{i−1}}. Further, for each such α, length[f^{m}(α)] is bounded by a polynomial in m, or it grows as (const)λ_{i}^{m} for some Perron-Frobenius number λ_{i} > 1 uniformly bounded away from 1. (f grows as g if f(m)/g(m) → 1 as m → ∞). Moreover, if such exponentially growing paths don’t exist, then for every edge e in \overline{G_{i} \ G_{i−1}} the edge-path f(e) crosses exactly one edge of \overline{G_{i} \ G_{i−1}}, and it crosses it once. Proofs are in [BH, section 5].

§1. Stable and Unstable Laminations

In this section we define the notion of the (un)stable lamination associated to an irreducible outer automorphism. More generally, in [BFH1] we associate finitely many laminations to any outer automorphism. A version of this construction was considered by Lustig [Lu1].

Let g : G → G be a train-track map with an irreducible transition matrix. We endow G with the structure of a marked ℝ-graph so that g expands lengths of edges by a uniform factor λ > 1. Let x ∈ G be any g-periodic point in the interior of some edge. Let ϵ > 0 be small, and let U be the ϵ-neighborhood of x. Then for some N > 0 we have f^{N}(U) ⊃ U. Choose an isometry ℓ : (−ϵ, ϵ) → U and extend it to the (unique) locally isometric immersion ℓ : ℝ → G such that ℓ(λ^{N}t) = g^{N}(ℓ(t)). We say that ℓ is obtained by iterating a neighborhood of x.

Definition 1.1. Two isometric immersions [a, b] → G and [c, d] → G are equivalent if there is an isometry [a, b] → [c, d] making the triangle commute.

If P is an equivalence class represented by γ : [a, b] → G, define g(P) to be the equivalence class of gγ pulled tight and scaled so it is an isometric immersion.

A leaf segment of an isometric immersion ℝ → G is the equivalence class of the restriction to a finite interval.

Two isometric immersions ℓ, ℓ′ : ℝ → G are (weakly) equivalent if every leaf segment of ℓ is also a leaf segment of ℓ′ and vice versa.

Lemma 1.2. Suppose x′ ∈ G is another g-periodic point in the interior of some edge of G, and ℓ′ : ℝ → G is obtained by iterating a neighborhood of x′. Then ℓ and ℓ′ are equivalent.

Proof. We show that any leaf segment of ℓ is also a leaf segment of ℓ′. The converse is similar.

Since the transition matrix of g is irreducible, ℓ′ is surjective. Choose an isometry ψ : (−ϵ, ϵ) → (a − ϵ, a + ϵ) ⊂ ℝ such that ℓ = ℓ′ψ on (−ϵ, ϵ). Say ℓ′(λ^{N′}t) = g^{N′}(ℓ′(t)). For any integer k > 0 and any t ∈ U we have

ℓ(λ^{kNN′}t) = g^{kNN′}(ℓ(t)) = g^{kNN′}(ℓ′ψ(t)) = ℓ′(λ^{kNN′}ψ(t)).

The claim now follows from the observation that for sufficiently large k the segment λ^{kNN′}(−ϵ, ϵ) contains any prechosen interval. □

Definition 1.3. The stable lamination Λ = Λ_{g}^{+}(G) (in G-coordinates) associated to g : G → G is the equivalence class of isometric immersions containing some (any) immersion obtained by iterating a neighborhood of a periodic point as above. A leaf of Λ is any immersion representing Λ. A leaf segment of Λ is a leaf segment of some (any) leaf ℓ of Λ.

Remarks 1.4. Any edge of G is a leaf segment (of Λ). Any g-iterate of a leaf segment is a leaf segment. Any subsegment of a leaf segment is a leaf segment. Any leaf segment is a subsegment of a sufficiently high iterate of an edge. For any leaf segment P there is a leaf segment P′ such that P = g(P′).

Let α be any immersed loop in G, and let k be the number of edges it crosses (counted with multiplicities). Then any g-iterate of α (pulled tight) can be written as a concatenation of ≤ k leaf segments.

Definition 1.5. We say that a sequence {α_{i}} of isometric immersions S_{i}^{1} → G (where the metric on S_{i}^{1} is a scalar multiple of the standard path metric, and the scale depends on i) weakly converges to Λ if the following holds (by m denote scaled Lebesgue measure):

For every L > 0 the ratio

\frac{m(\{x ∈ S_{i}^{1}|the L − nbhd of x is a leafsegment\})}{m(S_{i}^{1})}

converges to 1 as i → ∞.

Proposition 1.6. Suppose that α is an immersed loop in G representing a gnonperiodic conjugacy class. Then the sequence [g^{i}(α)] of tightenings of g^{i}(α) weakly converges to Λ.

Proof. This follows directly from the last paragraph in Remark 1.4. □

Definition 1.7. An isometric immersion ℓ : ℝ → G is quasiperiodic if for every L > 0 there exists L′ > L such that every leaf segment of ℓ of length L occurs as a subleaf segment of any leaf segment of ℓ of length L′.

Proposition 1.8. Every leaf of Λ_{g}^{+}(G) is quasiperiodic.

Proof. We first verify that any leaf segment Π that consists of only two edges of G occurs as a subleaf segment of any sufficiently long leaf segment.

Indeed, there is a constant L_{0} > 0 so that if a leaf segment P has length ≥ L_{0}, then it contains a subleaf segment which is an edge of G. There is N such that g^{N} restricted to any edge crosses all turns crossed by leaves of Λ_{g}^{+}(G), so Π is a subleaf segment of g^{N}(P). Therefore, if P′ is any leaf segment of length ≥ λ^{N}L_{0}, then P′ = g^{N}(P) for a leaf segment P of length ≥ L_{0}, and Π is a subsegment of P′.

Now let L > 0 be given. There is an integer M > 0 such that any leaf segment of length ≤ λ^{−M}L is a subsegment of a two-edge leaf segment considered above. Now set L′ = λ^{M+N}L_{0}.

Let P be a leaf segment of length L and P′ a leaf segment of length L′. Then P = g^{M}(Π) for a leaf segment Π contained in a two-edge leaf segment, and P′ = g^{M}(Π′) for some leaf segment Π′ of length λ^{N}L_{0}. By the special case, Π is a subsegment of Π′, and hence P is a subsegment of P′. □

Changing Coordinates. Suppose that g : G → G and Λ = Λ_{g}^{+}(G) are as above, and let H be any other marked ℝ-graph of the same rank. By τ : G → H denote the homotopy equivalence corresponding to the difference of the two markings. For any isometric immersion ℓ : ℝ → G denote by τ(ℓ) the unique (up to precomposition by an isometry of ℝ) isometric immersion ℝ → H proper homotopy equivalent to τg.

Lemma 1.9.

(1) If ℓ, ℓ′ : ℝ → G are equivalent, then τ(ℓ), τ(ℓ′) are also equivalent.

(2) If ℓ is quasiperiodic, so is τ(ℓ).

Proof. τ can be factored as the composition of a homeomorphism (which changes the metric on each edge) and a finite sequence of folds (see [St]). That the lemma is true for homeomorphisms and folds is left as an exercise. □

Definition 1.10. The stable lamination of g : G → G in the H-coordinates is the equivalence class Λ_{g}^{+}(H) containing τ(ℓ) for some (any) leaf ℓ of Λ_{g}^{+}(G).

The method of factoring through homeomorphisms and folds combined with Proposition 1.6 also establishes the following.

Lemma 1.11. Let α be a loop in G representing a g-nonperiodic conjugacy class. Then the sequence {[τ(g^{i}(α))]} weakly converges to Λ_{g}^{+}(H). □

Lemma 1.12. Suppose h : H → H is a train-track map representing the same outer automorphism of F_{n} as g : G → G, i.e. τg ∼ hτ. Then Λ_{g}^{+}(H) = Λ_{h}^{+}(H).

Proof. Let α be a loop as in Lemma 1.11. The sequence [τ(g^{i}(α))] = [h^{i}(τ(α))] weakly converges to Λ_{h}^{+}(H) (by Proposition 1.6) and to Λ_{g}^{+}(H) (by Lemma 1.11).

Consequently, Λ_{g}^{+}(H) and Λ_{h}^{+}(H) have arbitrarily long common leaf segments. Since they are both quasiperiodic, it follows that their leaves are equivalent, so that Λ_{g}^{+}(H) = Λ_{h}^{+}(H).

Definition 1.13. The stable lamination Λ_{ϕ}^{+} of an irreducible outer automorphism ϕ of F_{n} is the collection {Λ_{g}^{+}(H)|H is a marked ℝ − graph} where g : G → G is a train-track representative of ϕ. The unstable lamination Λ_{ϕ}^{−} of ϕ is the stable lamination of ϕ^{−1}.

Remark 1.14. Lemma 1.12 implies that the definition does not depend on the choice of a train-track representative. A marking of an ℝ-graph G of rank n gives a way of identifying the ends of the universal cover G̃ with the ends ∂F_{n} of F_{n}. Each leaf of a lamination corresponds to two ends of F_{n}. A coordinate free definition of the stable lamination as a subset of ∂F_{n} × ∂F_{n} can also be given.

§2. Out(F_{n})-action on the set of laminations

We denote by IL = IL(n) the set of stable laminations Λ_{ϕ}^{+} as ϕ ranges over all irreducible outer automorphisms of F_{n}. The group Out(F_{n}) acts on IL via

ψΛ_{ϕ}^{+} = Λ_{ψϕψ^{−1}}^{+}.

Explicitly, if ℓ is a leaf of Λ_{ϕ}^{+} in the H-coordinates and f : H → H a homotopy equivalence representing ψ then [f(ℓ)] represents a leaf of Λ_{ψϕψ^{−1}}^{+}.

Clearly, the infinite cyclic group generated by ϕ fixes Λ_{ϕ}^{+}. In the process of proving the Main Theorem, an essential step is to show that the stabilizer of Λ_{ϕ}^{+} is virtually cyclic. In particular, we have to show that if Λ_{ϕ}^{+} is realized in Hcoordinates, all leaves are surjective (so that Λ_{ϕ}^{+} is not carried by a free factor). Our Proposition 2.4 establishes a stronger result, and is preceded by a more technical lemma.

Let g : G → G be a train-track realization of an irreducible outer automorphism ϕ. We equip G with a minimal train-track structure; namely, we declare a turn legal iff it is crossed by leaves of Λ_{g}^{+}(G). Equivalently, a turn is legal iff there is a g-iterate of an edge of G that crosses the turn. We note the following two properties of g : G → G.

• The transition matrix of g is irreducible.

• At every vertex of G the “Whitehead graph” of Λ_{g}^{+}(G) is connected. The Whitehead graph at v is formed from Lk(v, G) by connecting two points iff the corresponding turn is legal.

If the latter property fails we can construct a reduction of ϕ by blowing up G as in [BH, Proposition 4.5].

A technical difficulty arises since (unlike for surface homeomorphisms) a finitesheeted covering of g : G → G might represent a reducible automorphism. For example, g could correspond to a pseudo-Anosov homeomorphism on a once punctured surface, and a double cover g̃ of g could correspond to a pseudo-Anosov homeomorphism of a twice punctured surface fixing the punctures. As an automorphism of a free group, g̃ is reducible (each puncture represents a fixed basis element). However, the following weaker form of irreducibility still holds.

Lemma 2.1. Suppose π : G′ → G is a finite-sheeted covering space and g′ : G′ → G′ a lift of g. Then g′ satisfies

(1) the transition matrix of g′ is irreducible (so in particular the legal structure on the turns in G′ and the Whitehead graph of g′ at a vertex of G′ are well-defined) and

(2) the Whitehead graph of g′ at a vertex v of G′ is the lift of the Whitehead graph of g at π(v), and in particular it is connected.

Proof. Lift the legal turn structure from G to G′. We will show that this structure is minimal. Since the Whitehead graphs at all vertices are connected, it follows that any two edges of G′ can be connected by a legal path (=path that makes only legal turns).

Let a′ and b′ be two edges in G′ such that a′b′ is legal. We claim that some g′-iterate of a′ contains … a′b′ . . ..

Let a and b be the projections of a′ and b′ respectively. By passing to a power, we may assume that g(a) = … ab . . .. This generates a fixed point in a, call it x. Since g is a homotopy equivalence, g′ must permute the finite set π^{−1}(x). By passing to a further power, we may assume that g′ fixes this set pointwise. In particular, we have a fixed point x′ ∈ a′, and g′ expands a′ across b′.

A similar argument shows that some iterate of b′ crosses a′b′.

We next argue that the transition matrix for g′ is irreducible. In light of the above discussion, it suffices to show that for any two edges a, b in G′ there is sequence a = e_{0}, e_{1}, ⋯, e_{k} = b of edges in G′ such that consecutive edges are adjacent and form a legal turn (with respect to the lifted structure). Let H′ be the maximal subgraph of G′ consisting of edges b that admit such a sequence (with a fixed a). Then H′ is a nonempty subgraph. If H′ ≠ G′, then there is a vertex v in G′ that is incident to edges both in H′ and in G′ \ H. The Whitehead graph at v (with respect to the lifted structure) is connected, so in particular an edge in H′ and an edge in G′ \ H′ form a legal turn at v, contradicting the maximality of H′.

The lemma follows. □

Definition 2.2. We say that (the conjugacy class of) a subgroup A ⊂ F_{n} carries Λ_{ϕ}^{+} if there is a marked ℝ-graph H, an isometric immersion τ : Γ → H of graphs with Im π_{1}(τ) = A, and an isometric immersion ℓ : ℝ → Γ such that τℓ is a leaf of Λ_{ϕ}^{+}(H).

Remark 2.3. If one leaf lifts to Γ, so does every leaf. If we find a factorization as above using a marked ℝ-graph H, then we can find a similar factorization using any prechosen marked ℝ-graph.

Proposition 2.4. If a finitely generated subgroup A ⊂ F_{n} carries Λ_{ϕ}^{+}, then A has finite index in F_{n}.

Proof. Let g : G → G be a train-track realization of ϕ, and let Γ → G be an isometric immersion corresponding to A ⊂ F_{n} . Then Γ is a finite graph. Complete the immersion by adding vertices and edges to Γ to a connected finite-sheeted covering space G′ → G (see [St]). Notice that if A has infinite index, then we really are adding new edges, and hence a lift of a leaf does not cross every edge of G′. Now pass to a further finite cover to which g lifts. But now the leaves of the stable lamination of the lift don’t cross those edges that cover added edges, contradicting the lemma. □

Examples 2.5.

(1) Λ cannot be carried by a proper free factor of F_{n}.

(2) If h : H → H is a relative train-track map, then there is an immersion [BH, section 6] H′ → H with H′ a finite graph such that all periodic Nielsen paths in H lift to H′ and all loops in H′ map to h-periodic loops. Then leaves of Λ do not lift to H′, unless h is periodic. A concrete example: Let H be the bouquet of 3 circles corresponding to a, b, and c. For h : H → H given by h(a) = ab, h(b) = bab, h(c) = c[a, b] we can take H′ to be two circles joined by an arc representing F ix(π_{1}(h)) =< [a, b], c[a, b]c^{−1} >.

Fix an irreducible outer automorphism ϕ, a train-track representative g : G → G, and let Λ = Λ_{ϕ}^{+}. We next study the stabilizer Stab(Λ).

Proposition 2.6. Let ψ ∈ Stab(Λ), and let h : H → H be a relative train-track representative of ψ. Then

(1) h has finite order on every proper h-invariant subgraph of H without valence 1 vertices that is a union of strata, and

(2) if h has no exponentially growing strata, then h has finite order.

Proof. (1) Let ℓ : ℝ → H represent a leaf of Λ in the H-coordinates and let H_{0} ⊂ H be a proper h-invariant subgraph. By Proposition 2.4 ℓ is a concatenation of nondegenerate segments in H_{0} and in H \ H_{0} (otherwise ℓ would “lift” to a proper subgraph of H). Notice that all H_{0}-segments are Nielsen (periodic) or else h-iteration will produce arbitrarily long leaf segments contained in H_{0} contradicting quasiperiodicity. Again by quasiperiodicity, there is an upper bound to the length of both H_{0} and H \ H_{0}-segments, and hence only finitely many segments occur. Start with the disjoint union X of copies of the segments and the natural immersion X → H. Identify two endpoints of X if they are mapped to the same point of H. Then fold to convert the resulting map to an immersion π : X′ → H. Since ℓ lifts to X′ by construction, π must be a finite-sheeted covering space (by Proposition 2.4). Therefore, a power of any loop in H_{0} lifts to X′. It follows that this power is a concatenation of paths in H_{0} each of which has finite h-order. Thus every loop in H_{0} has finite h-order and hence h|H_{0} has finite order by [BH].

(2) By (1), we are reduced to analyzing the case where H_{0} ⊂ H is an h-invariant subgraph, h|H_{0} has finite order, and the transition matrix of the block H \ H_{0} is a permutation matrix. By passing to a power, we may assume that h|H_{0} = id, that H_{0} is a bouquet of circles, or the disjoint union of two bouquets of circles, that c is the only edge in H \ H_{0}, and h(c) = v^{k}cw^{l}, where v and w are primitive loops in H_{0}. If both v and w are nontrivial, then a leaf of Λ (in H-coordinates) must look like

… cw^{a_{−1}}c^{−1}v^{b_{−1}}cw^{a_{0}}c^{−1}v^{b_{0}}cw^{a_{1}}c^{−1}v^{b_{1}}c …

for some integers a_{i}, b_{i}, or else iteration produces arbitrarily large leaf segments in H_{0}. Thus, leaves of Λ lift to the graph consisting of two disjoint loops corresponding to v and w, and they are connected by an edge corresponding to c. Therefore, this graph is a finite-sheeted covering space of H. Since c has only one lift, it is a homeomorphism, and h is homotopic to id.

If v is trivial, but w nontrivial, then a leaf of Λ must look like

… cw^{a_{−1}}c^{−1}x_{−1}cw^{a_{0}}c^{−1}xcw^{a_{1}}c^{−1}x_{1}c …

for some integers a_{i} and H_{0}-paths x_{i}. In this case, Λ lifts to the graph consisting of the disjoint union of H_{0} and a loop corresponding to w joined by an edge corresponding to c. Again, this graph must be homeomorphic to H, contradicting the fact that w has two lifts.

The case where w is trivial and v nontrivial is similar. □

The λ-map.

Lemma 2.7. Suppose h : G → G represents an outer automorphism ψ. Then there is a positive number λ = λ(h, Λ) such that for every ϵ > 0 there is N > 0 so that if L is a leaf segment of Λ of length > N, then

|\frac{length([h(L)])}{length(L)} − λ| < ϵ

(all lengths are measured in G).

Proof. Perron-Frobenius Theorem applied to the transition matrix of g : G → G shows that long leaf segments of Λ cross edges of G with frequencies close to those determined by the components of the Perron-Frobenius eigenvector. Fix a large number K and note that (even longer) leaf segments are concatenations of leaf segments of the form g^{K}(e) for edges e of G, each occurring with definite frequency. (For K = 1 this is just the preceding sentence. For K > 1 apply g^{K}.) If N is large, we can think of L as being a concatenation of such leaf segments (short segments near the ends of L can be discarded as their contribution is negligible). Let C denote the cancellation constant for h : G → G. Denote by l_{e} the length of g^{K}(e), by l_{e}^{h} the length of [h(g^{K}(e))], and by N_{e} the number of occurrences of g^{K}(e) in L. Then

length(L) = ∑_{e} N_{e}l_{e}

and

∑_{e} N_{e}(l_{e}^{h} − 2C) ≤ length([h(L)]) ≤ ∑_{e} N_{e}l_{e}^{h}

It follows that

\begin{matrix}∑ & e & N_{e}l_{e}^{h} \\ ∑ & e & N_{e}l_{e} \end{matrix} − 2C \frac{∑ N_{e}}{∑ N_{e}l_{e}} ≤ \frac{length(h(L))}{length(L)} ≤ \begin{matrix}∑ & e & N_{e}l_{e}^{h} \\ ∑ & e & N_{e}l_{e} \end{matrix}

The error term 2C \frac{∑ N_{e}}{∑ N_{e}l_{e}} clearly goes to 0 as K → ∞. Since the frequencies \frac{N_{e}}{∑_{e} N_{e}} converge to the coordinates of the Perron-Frobenius eigenvector as length(L) → ∞, the claim follows. □

Note that the above discussion does not imply that λ is algebraic. Now define σ : Stab(Λ) → ℝ^{+} by σ(ψ) = λ(h, Λ) for a representative h : G → G of ψ. It is then clear that σ is a homomorphism (the image by h of a long segment of Λ is again a long segment of Λ except for a bounded error near the endpoints).

We next study the kernel and the image of σ.

Lemma 2.8. If ψ ∈ Stab(Λ) is exponentially growing and reducible, then σ(h) is bounded away from 1. In particular, there is no exponentially growing reducible ψ ∈ Ker(σ).

Proof. By Proposition 2.6(1), we may assume that the relative train-track map h : H → H representing ψ has finite order on a subgraph H_{0} and has an exponentially growing block H \ H_{0}. But then it is easy to see, by considering the subgraphovergraph decomposition of any leaf of Λ in H-coordinates, that lengths of leaves grow exponentially, at the rate equal to the growth rate of the transition matrix of h. (Not all overgraph leaf segments can be Nielsen.) □

We now aim to generalize Lemma 2.8 to irreducible automorphisms.

Lemma 2.9. Let h : H → H be a train-track map representing an irreducible outer automorphism ρ. Then for every C > 0 there is a number M > 0 such that if L is any path, then one of the following holds.

(a) [h^{M}(L)] contains a legal segment of length > C.

(b) [h^{M}(L)] has fewer illegal turns than L.

(c) L is a concatenation x · y · z with [h^{M}(y)] Nielsen and x and z have length ≤ 2C and at most one illegal turn.

Proof. Choose M so that it exceeds the number of legal edge-paths of length ≤ 2C. Now assume that L is a path such that (b) fails. Then iteration of L amounts to iterating maximal legal subsegments of L and canceling portions of adjacent ones. If, in addition, (a) fails as well, then each maximal legal segment of L, except possibly the ones that contain endpoints, must have two iterates that after cancellation yield the same segment. In particular, each such segment contains a preperiodic point so that these points subdivide L as x · y_{1} · y_{2} ⋯ y_{l} · z, and we have (c). □

Lemma 2.10. Let ρ be a nongeometric irreducible automorphism (and thus its inverse is also nongeometric), let h : H → H and h′ : H′ → H′ be train-track representatives of ρ and ρ^{−1} respectively, and let τ : H → H′ and τ′ : H′ → H be Lipschitz homotopy equivalences corresponding to differences of markings. Then for any C > 0 there are constants N_{0} > 0 and L_{0} such that if ȷ is an isometric immersion into H of the real line, a segment of length ≥ L_{0}, or a circle of length ≥ L_{0}, and if ȷ′ is the isometric immersion obtained from τȷ by pulling tight, then one of the following holds.

(A) [h^{N_{0}}ȷ] contains a legal segment of length > C.

(B) [h^{′N_{0}}ȷ′] contains a legal segment of length > C.

Proof. There is an integer m such that it is impossible to concatenate more than m Nielsen paths in H or H′. Without loss, we may assume that C is larger than the critical constants for h and h′. Let M be the larger of the two integers guaranteed by Lemma 2.9 applied to h, C and h′, C. We will fix a large integer s = s(h, h′, τ, τ′, M) (to be specified below). Suppose now that (A) does not hold with N_{0} = sM. We will apply Lemma 2.9 only to segments L ⊂ ȷ such that h^{M}(∂L) ⊂ [h^{M}(ȷ)]. Call such segments h^{M}-admissible. By our assumption, possibility (a) never occurs. If we further restrict to segments L with > m + 2 illegal turns, then (c) cannot hold either. Thus for such segments (b) always holds. We can represent ȷ as a concatenation of such segments of uniformly bounded length, and the uniform bound does not depend on ȷ, but only on h, h′, τ, τ′, M.

Say p is an upper bound to the number of illegal turns in each segment. Fix α with \frac{p−1}{p} < α < 1. For long enough segments L in ȷ the ratio

\frac{number of illegal turns in [h^{M}L]}{number of illegal turns in L}

is bounded above by α.

Applying the same construction to h^{M}(ȷ), and then to h^{2M}(ȷ) etc. we see that for given s > 0 and long enough segments L ⊂ ȷ (the length depends on s as well)

\frac{number of illegal turns in [h^{sM}L]}{number of illegal turns in L} < α^{s},

or else (A) holds with N_{0} = sM. Since legal segments have length bounded above by C and below by the length of the shortest edge (with the exception of the two containing the endpoints), the length is comparable to the number of illegal turns. Therefore, if (A) fails,

\frac{length_{H}[h^{sM}L]}{length_{H}(L)} < const(h, C)α^{s}

Repeat the discussion with [τh^{sM}ȷ] in place of ȷ, and with h′ in place of h. If (B) fails as well (with N_{0} = sM), we reach a similar conclusion, that

\frac{length_{H′}[h^{′sM}τh^{sM}L]}{length_{H′}[τh^{sM}L]} < const(h′, C)α^{s}.

Multiplying, observing that h^{′sM}τh^{sM} ∼ τ′ (and the homotopy is bounded so that for long L the ratio

\frac{length_{H′}[h^{′sM}τh^{sM}L]}{length_{H′}[τ′(L)]}

is in the interval [\frac{1}{2}, 2]), and rearranging terms we obtain

\frac{1}{2}Lip(τ)Lip(τ′) ≤ \frac{length_{H}[h^{sM}L]}{length_{H′}[τh^{sM}L]} \frac{length_{H′}[τ′L]}{length_{H}(L)} < 2const(h)const(h′)α^{2s}

where Lip(·) denotes the Lipschitz constant. For large s we reach a contradiction. □

Definition 2.11. We say that a sequence {Λ_{i}} of irreducible laminations in IL weakly converges to Λ ∈ IL if for some (any) marked ℝ-graph H every leaf segment of Λ in H-coordinates is a leaf segment of Λ_{i} in H-coordinates for all but finitely many i.

Proposition 2.12. Let Λ = Λ_{ϕ}^{+} ∈ IL be an irreducible lamination, and let ψ, along with ϕ, be an irreducible outer automorphism of F_{n}. Then either the forward ψ-iterates of Λ weakly converge to Λ_{ψ}^{+} or Λ = Λ_{ψ}^{−}. In particular, if ψ ∈ Stab(Λ), then Λ = Λ_{ψ}^{±}.

Proof. First consider the case that ψ is nongeometric. We adopt the notation from Lemma 2.10. Let ℓ be a leaf of Λ in the H-coordinates. We apply Lemma 2.10 to [h^{K}ℓ] with K > 0 and C larger than critical constants of h and h′. If for some K > 0 (A) holds, then it follows from quasiperiodicity that the forward iterates weakly converge to Λ_{ψ}^{+}. The remaining possibility is that [τh^{K}ℓ] contains an H′- legal segment of length > C for all K > 0. But this means that [τℓ], which equals [h^{′K}τh^{K}ℓ] up to bounded error, contains an arbitrarily high h′-iterate of a legal segment, hence Λ = Λ_{h}^{−} (again by quasiperiodicity).

When ψ is geometric, we appeal to standard surface theory. There is a realization of ψ as a pseudoAnosov homeomorphism of a punctured surface F. A leaf ℓ of Λ can be realized as a biinfinite path in F. If ℓ is not properly homotopic to a leaf of Λ_{ψ}^{−}, then it essentially crosses its leaves. A suitable lift of ψ to the universal covering space acts on the circle at infinity and the endpoints of a lift of ℓ converge under forward iteration to attracting periodic orbits. Furthermore, the lift of ℓ converges (after pulling tight) to the orbit determined by one of sides or diagonals of the polygon whose vertices are attracting periodic points. For more details, consult [Th],[HT],[CB], or [FLP]. □

Corollary 2.13. Lemma 2.8 holds without assuming that ψ is reducible. □

Theorem 2.14. For every irreducible ϕ the stabilizer Stab(Λ_{ϕ}^{+}) is virtually cyclic.

Proof. The map σ : Stab(Λ_{ϕ}^{+}) → ℝ^{+} has finite kernel and discrete image. (Recall [CV] that any torsion subgroup of Out(F_{n}) is finite.) □

Corollary 2.15. If ψ ∈ Stab(Λ_{ϕ}^{+}) has σ(ψ) ≠ 1, then ψ and ϕ have common nonzero powers.

Proof. There are infinitely many pairs a, b of integers with σ(ϕ^{a}) = σ(ψ^{b}). Since the kernel of σ is finite, we get equations of the form

ϕ^{a}ψ^{−b} = ϕ^{c}ψ^{−d}

which implies the claim. □

Proposition 2.16. Let ϕ and ψ be two irreducibles. The following statements are equivalent.

(1) {Λ_{ϕ}^{±}} = {Λ_{ψ}^{±}}.

(2) {Λ_{ϕ}^{±}} ∩ {Λ_{ψ}^{±}} ≠ ∅.

(3) ϕ and ψ have common nonzero powers.

Proof. That (3) implies (1) which implies (2) is obvious. (3) follows from (2) by Corollary 2.15. □

§3. Trees

In this section we examine the relationship between laminations and trees. We assume that the reader is familiar with basic results in the theory of ℝ-trees. See [CM]. All of the trees appearing in this section will be limits of free simplicial trees. Let X denote the closure of Culler-Vogtmann’s Outer Space and let X̂ denote the space of trees representing elements of X. Although we will not need it, we note that X̂ is the space of very small trees [BF1].

The Bounded Cancellation Lemma. We need the following two lemmas. The first one generalizes Cooper’s Bounded Cancellation Lemma [Co].

Lemma 3.1. Let f : T_{0} → T be an F_{n}-map from a free simplicial tree T_{0} to a tree T ∈ X̂. Then

BCC(f) ≤ Lip(f)vol(T_{0})

where vol(T_{0}) is the sum of lengths of edges of T′/F_{n}.

Proof. By scaling the metric on T_{0} we may assume Lip(f) = 1. After redefining the map on edges (rel vertices) and decreasing vol(T_{0}) we may assume that f isometrically embeds every edge. First restrict to the case where T is also free simplicial. In that case we prove a sharper inequality

BCC(f) ≤ vol(T_{0}) − vol(T)

If f_{1} : T_{1} → T_{2} and f_{2} : T_{2} → T_{3} are two such maps, then BCC(f_{2}f_{1}) ≤ BCC(f_{1})+ BCC(f_{2}) so if the sharper inequality holds for f_{1} and f_{2}, it also holds for f_{2}f_{1}. The proof in this special case now follows from the fact that any f as above can be factored as a finite composition of folds, and for a fold the inequality is evident.

If there is f : T_{0} → T (with T in X̂) for which BCC(f) > Lip(f)vol(T_{0}), then there is a free simplicial T′′ approximating T and an F_{n}-map f′ : T_{0} → T′′ approximating f with Lip(f′) close to Lip(f), and BCC(f′) > BCC(f)−ϵ. Hence the inequality is violated for f′. □

Lemma 3.2. [Sk2][Wh]

(1) There is a (continuous) section Σ : X → X̂.

(2) Let T_{0} ∈ X̂ be free and simplicial. For every T ∈ Σ(X) there is an F_{n}-map f_{T_{0},T} : T_{0} → T that varies continuously with T and so that the Lipschitz constants Lip(f_{T_{0},T}) are uniformly bounded.

Laminations vs. Trees. Let Λ be the stable lamination of some irreducible outer automorphism.

Definition 3.3. Let H be a marked ℝ-graph of rank n. We say that the length of Λ in a tree T is 0, and write l_{T} (Λ) = 0, if there is a constant C = C(H, T) such that the image of every segment of Λ under f_{Ĥ,T} has diameter ≤ C. More precisely, the condition is that if ℓ : ℝ → H is a leaf of Λ and if ℓ̃: ℝ → H̃ is a lift of ℓ, then for every segment L ⊂ ℝ the of [f_{Ĥ,T} (L)] is ≤ C.

Clearly, this depends only on the projective class of T and not on the choice of H or f_{Ĥ,T}. Also note that if T is free simplicial, then the length of Λ in T is not 0.

Recall that Out(F_{n}) acts on X̂ on the right via

l_{(T)ϕ}(γ) = l_{T} (ϕ(γ)).

This action induces an action of Out(F_{n}) on X. The stable tree T^{+} = T_{ϕ}^{+} of ϕ is defined to be the limit of the sequence {(G̃)ϕ^{i}} for a train-track g : G → G for ϕ. If λ is the growth rate of g, then T^{+} is the projective class of the tree T̂^{+} given by

lT̂^{+}(γ) = lim_{i→∞} \frac{l_{H}(g^{i}(γ))}{λ^{i}}

and in particular

lT̂^{+}(g(γ)) = λlT̂^{+}(γ)

The unstable tree T^{−} = T_{ϕ}^{−} is defined to be the stable tree of ϕ^{−1}. That these definitions don’t depend on the choice of a train-track follows from the next Lemma.

Lemma 3.4. If the length of Λ_{ϕ}^{+} in T ∈ X is not 0, then T converges to T_{ϕ}^{+} under iteration by ϕ. In particular, this applies to all free simplicial trees T. Further, the convergence is uniform on compact sets K ⊂ X where Λ_{ϕ}^{+} has nonzero length.

Proof. Let g : G → G be a train-track representative of ϕ. If L is a sufficiently long leaf segment of Λ_{ϕ}^{+} (in the G-coordinates), then its image in T̂ is long. Let β be a legal loop. If α is a g-nonperiodic loop in G, then for large N both g^{N}(α) and g^{N}(β) can be viewed (up to bounded error) as concatenations of long segments that occur with approximately the same frequencies. Therefore

lim_{N→∞} \frac{lT̂(ϕ^{N}(α))}{lT̂(ϕ^{N}(β))}

is independent of T̂ ∈ Σ(K). Since l_{T} is constant on conjugacy classes, the slight abuse in the last equation is justified. If α is periodic, the above limit is again independent of T̂ ∈ Σ(K) (and equal to 0). The claim now follows by observing that we may assume that T_{ϕ}^{+} ∈ K. The details of this argument are similar to those of the proof of Lemma 2.7. □

Lemma 3.5. Let g : G → G be a train-track representative for ϕ. Let Λ be a stable lamination (perhaps unrelated to ϕ).

(1) For T ∈ X, if there is a leaf segment of Λ (in the G-coordinates) whose image in Σ(T) under f_{G̃,Σ(T)} (pulled tight) has length > 2BCC(f_{G̃,Σ(T)}), then Λ does not have length 0 in T.

(2) The set of trees T ∈ X with l_{T} (Λ) = 0 is closed.

(3) l_{T_{ϕ}^{+}}(Λ_{ϕ}^{−}) = 0, and similarly l_{T_{ϕ}^{−}}(Λ_{ϕ}^{+}) = 0.

(4) If Λ ≠ Λ_{ϕ}^{−}, then Λ does not have length 0 in T_{ϕ}^{+}.

Proof. (1) This follows from quasiperiodicity of Λ. If L is a leaf segment of Λ whose image in Σ(T) has length BCC(f_{G̃,Σ(T)}) + ϵ, then a leaf segment of Λ that contains N disjoint copies of L maps to a segment (pulled tight) in Σ(T) of length ≥ 2BCC(f_{G̃,Σ(T)}) + Nϵ.

(2) This is a consequence of (1). If T′ is a tree such that Λ has nonzero length in T′, then there is a leaf segment L in a leaf of Λ that maps to a segment in Σ(T′) of length greater than 2BCC(f_{G̃,Σ(T)}) for all T ∈ X. In particular, the image of L in Σ(T) is longer than 2BCC(f_{G̃,Σ(T)}) for all T in a neighborhood of T′.

(3) This is easy to see directly. Let γ be a loop in G representing a ϕ-nonperiodic element. Let l be the length of γ in Σ(T_{ϕ}^{−}). Then the length of ϕ^{N}(γ) in Σ(T_{ϕ}^{−}) is lλ^{−N}, which goes to 0 as N → ∞. But for large N the loop ϕ^{N}(γ) will contain long leaf segments of Λ_{ϕ}^{+}. These leaf segments map in Σ(T_{ϕ}^{−}) to segments of length ≤ 2BCC(f_{G̃,Σ(T_{ϕ}^{−})}) by an argument as in (1).

(4) If Λ has length 0 in T_{ϕ}^{+}, then all iterates of Λ by ϕ have length 0 in T_{ϕ}^{+}. It follows that under forward iteration by ϕ, Λ couldn’t converge to Λ_{ϕ}^{+} (or else some positive iterate would contain a leaf segment that maps to a long segment in Σ(T_{ϕ}^{+})). By Proposition 2.12, Λ = Λ_{ϕ}^{−}. □

Now denote by IT = IT (n) the set of irreducible trees, that is trees of the form T_{ϕ}^{+} for some irreducible outer automorphism ϕ of F_{n}.

Corollary 3.6. The function Φ : IT → IL that takes an irreducible tree T = T_{ϕ}^{+} to the unique irreducible lamination Λ_{ϕ}^{+} that has length 0 in T is an Out(F_{n})- equivariant bijection. In particular, Stab(T) = Stab(Φ(T)), and Corollary 2.16 can be stated using trees instead of laminations.

Proof. Clearly, Φ is Out(F_{n})-equivariant and surjective. If Φ(T_{ϕ}^{+}) = Φ(T_{ψ}^{+}), then both ϕ and ψ stabilize the same irreducible lamination, and hence have common powers. It follows that T_{ϕ}^{+} = T_{ψ}^{+}, and Φ is injective. □

Proposition 3.7. Suppose that ϕ and ψ are two irreducibles that don’t have common powers. Then high powers of ϕ and ψ freely generate F_{2}.

Proof. By Proposition 2.16 and Corollary 3.6, the assumption on ϕ and ψ is equivalent to saying that the four trees T_{ϕ}^{±} and T_{ψ}^{±} are all distinct, and similarly the four laminations Λ_{ϕ}^{±} and Λ_{ψ}^{±} are all distinct. By Lemma 3.5(4) the only lamination from these four that has length 0 in T_{ϕ}^{+} is Λ_{ϕ}^{−}. (Three more analogous statements.) By Lemma 3.5(2) there are pairwise disjoint compact neighborhoods U_{ϕ}^{±} and U_{ψ}^{±} of T_{ϕ}^{±} and T_{ψ}^{±} respectively so that Λ_{ϕ}^{+} has nonzero length in every tree in U_{ψ}^{±} and U_{ϕ}^{+}. (Three more such statements.) By Lemma 3.4, there is N > 0 so that k ≥ N implies that ϕ^{k} maps U_{ψ}^{±} and U_{ϕ}^{+} into U_{ϕ}^{+} etc. Therefore we obtain the familiar ping-pong situation [Ti, section 3] implies the claim. □

§4. A Tits Alternative

In this section, we prove a special case of the Tits Alternative for Out(F_{n}), cf [BFH1-3].

Theorem 4.1. Suppose H is a is a subgroup of Out(F_{n}) that contains an irreducible outer automorphism ϕ of infinite order. Then either H contains F_{2} or H is virtually cyclic.

Proof. By Λ^{±} and T^{±} denote the stable and unstable laminations and trees of ϕ. We consider the following two cases.

Case 1. H fixes the set Λ^{±}. In this case, a subgroup of H of index 1 or 2 fixes Λ^{+}, hence H is virtually cyclic by Theorem 2.14.

Case 2. There is ρ ∈ H such that ρ({Λ^{±}}) ≠ {Λ^{±}}. Let ψ = ϕ^{ρ}. Then Λ_{ψ}^{±} = ρ(Λ^{±}) and the equivalent conditions of Corollary 2.16 fail. By Proposition 3.7 high powers of ϕ and ψ generate F_{2}. □

§5. Further Applications

We conclude this work with four other theorems that can be proved with the techniques developed here. The first of these, Theorem 5.1, was announced in [BF2] (for automorphisms, perhaps reducible, with no nontrivial periodic conjugacy classes – this more general statement may be proved using [BFH1]). Theorem 5.2 for the case of pseudo-Anosov homeomorphisms was established by L. Mosher [Mo]. Theorem 5.3 was also proved by M. Lustig [Lu2]. We assume throughout that h : H → H is a train-track representative of an irreducible outer automorphisms ϕ ∈ Out(F_{n}) and h′ : H′ → H′ represents ϕ^{−1}. By T^{+} and T^{−} denote the corresponding stable and unstable trees, and by Λ^{+} and Λ^{−} the corresponding laminations.

We also fix a basis of F_{n}. By ||α|| we denote the length of the conjugacy class α, i.e. the length of the cyclically reduced word representing α. For w ∈ F_{n} denote by |w| the word length of w.

Theorem 5.1. Assume that ϕ has no nontrivial periodic conjugacy classes (or equivalently that ϕ is nongeometric). Let ϕ̃: F_{n} → F_{n} be any lift of ϕ to Aut(F_{n}).

(1) There is an integer N_{1} such that for any conjugacy class α and any N ≥ N_{1}

3||α|| ≤ max(||ϕ^{N}(α)||, ||ϕ^{−N}(α)||).

(2) There is an integer N′_{1} such that for any w ∈ F_{n} and any N ≥ N′_{1}

2|w| ≤ max(|ϕ̃^{N}(w)|, |ϕ̃^{−N}(w)|).

(3) The semidirect product F_{n} ⋊ ℤ induced by ϕ̃ is word hyperbolic.

Theorem 5.2. Let ϕ and ψ be two irreducible outer automorphisms such that they don’t have common powers and neither has a nontrivial periodic conjugacy class. Let ϕ̃: F_{n} → F_{n} and ψ̃ : F_{n} → F_{n} be their arbitrary lifts to Aut(F_{n}).

(1) There is N_{2} > 0 such that for any conjugacy class γ and any N ≥ N_{2} at least 3 of the lengths ||ϕ^{N}(γ)||, ||ϕ^{−N}(γ)||, ||ψ^{N}(γ)||, ||ψ^{−N}(γ)|| are ≥ 3||γ||.

(2) There is N′_{2} such that for any w ∈ F_{n} and any N ≥ N′_{2} at least three of the lengths |ϕ̃^{N}(w)|, |ϕ̃^{−N}(w)|, |ψ̃^{N}(w)|, |ψ̃^{−N}(w)| are ≥ 2|w|.

(3) For all N ≥ N′_{2} the semidirect product F_{n} ⋊ F_{2} determined by ϕ̃^{N} and ψ̃^{N} is word hyperbolic.

Of course, constants 2 and 3 above are arbitrary.

Recall that T^{+} and T^{−} are the stable and the unstable trees for ϕ.

Theorem 5.3. The diagonal action of F_{n} on T^{+} × T^{−} is discrete.

Theorem 5.4. Let A be a finitely generated subgroup of F_{n} of infinite index. Then the action of A on T^{+} is discrete.

In the proofs we need several lemmas. Fix a constant C that is larger than the critical constant for h. For any immersed loop α in H, the legality of α is the ratio

LEG(α) = LEG_{H}(α) = ; \frac{sum of the lengths of legal leaf segments of α of length ≥ C}{length(α)}

Lemma 5.5. Assume ϕ is nongeometric.

(1) For every ϵ > 0 and A > 0 there is N_{3} = N_{3}(ϵ, A) such that if LEG_{H}(α) ≥ ϵ

then

length[h^{N}(α)] ≥ A length(α)

for all N ≥ N_{3}.

(2) For every ϵ > 0 there is δ > 0 such that if LEG_{H}(α) ≥ ϵ then l_{T^{−}}(α) ≥ δ.

(3) For every ϵ > 0 and every L > 0 there is N_{4} = N_{4}(ϵ, L) > 0 such that if LEG_{H}(α) ≥ ϵ, then for all N ≥ N_{4} the set of points of [h^{N}(α)] whose Lneighborhood is a leaf segment of Λ^{+} has total length ≥ (1−ϵ)length[h^{N}(α)]. In other words, the weak convergence towards Λ^{+} is uniform on the set LEG_{H} ≥ ϵ.

Proof. (1) Let μ > 0 be as in the introductory section. Then

\begin{matrix}length[h^{m}(α)] ≥ μλ^{m}length(legal leaf segments of α of length ≥ C) ≥ \\ μλ^{m}ϵ length(α) ≥ A length(α) \end{matrix}

when m is large enough.

(2) The above calculation shows l_{T^{−}}(α) ≥ μϵ length(α).

(3) α is a concatenation of ≤ b^{−1} length(α) edges (b is the length of the shortest edge in H), and thus [h^{N}(α)] is a concatenation of at most that many leaf segments. The set of points whose L-neighborhood doesn’t form a leaf segment have total length at most Lb^{−1} length(α), so we can take N_{4} = N_{3}(ϵ, L/bϵ). □

The following is, in essence, a restatement of Lemma 2.10. Let h′ : H′ → H′ be a train-track representative of ϕ^{−1}, and let τ, τ′ be as in Lemma 2.10. Assume C exceeds the critical constant of h′ as well. Let N_{0} be the integer from the conclusion of Lemma 2.10. When α is a conjugacy class, denote by α_{H}, α_{H′} the immersed loops in H, H′ representing α.

Lemma 5.6. Assume ϕ is nongeometric. Then there is ϵ > 0 such that for every nontrivial conjugacy class α and every N ≥ N_{0} either LEG_{H}(ϕ^{N}(α)_{H}) ≥ ϵ or LEG_{H′}(ϕ^{−N}(α)_{H′}) ≥ ϵ.

Proof. First note that Lemma 2.10 implies that one of ϕ^{N}(α)_{H} and ϕ^{−N}(α)_{H′} has a long legal segment. We must show that one of them has a definite fraction of long legal segments. Suppose not and let {α_{i}} be a sequence where these fractions converge to 0. Denote by τ : G → H and τ′ : G → H′ markings from the standard rose, and view the α′_{i}s as immersed loops in G. We can then find segments in α_{i} of arbitrarily long length that after applying τϕ^{N} and τ′ϕ^{−N} and tightening don’t contain legal segments of length ≥ C except within a bounded distance from the endpoints. Choosing a subsequence and limiting produces an immersed line ȷ in G whose τ-image in H violates Lemma 2.10. □

Corollary 5.7. Let {β_{i}} be a sequence of nontrivial conjugacy classes. Then one of the following holds.

(1) The sequence {ϕ^{−2i}(β_{i})} has a subsequence that weakly converges to Λ^{−}, or (2) The sequence {β_{i}} weakly converges to Λ^{+}.

Proof. Let N_{0} be as above and let ϵ be as in Lemma 5.6. Let α_{i} = ϕ^{−i}(β_{i}). If (1) fails then, by Lemma 5.5(3), LEG_{H′}(α_{i}) < ϵ for all but finitely many i. Thus by Lemma 5.6 we must have LEG_{H}(ϕ^{N_{0}}(α_{i})) ≥ ϵ for all large i. By Lemma 5.5(3) the collection {ϕ^{N_{0}}(α_{i})} uniformly weakly converges to Λ^{+} under forward iteration, and the claim follows. □

Proof of Theorem 5.1. (1) There is a constant K such that for any conjugacy class α the numbers ||α||, length_{H}(α_{H}), length_{H′}(α_{H′}) have ratios between 1/K and K. Let N_{0} be as above, and choose D > 0 such that for all nontrivial conjugacy classes α ||ϕ^{±N_{0}}(α)|| ≥ ||α||/D. Let ϵ be as in Lemma 5.6, and let N_{3} = N_{3}(ϵ, 3DK^{2}) be as in Lemma 5.4(1) for both h and h′. Then set N_{1} = N_{0} + N_{3}.

If α is any nontrivial conjugacy class, then by Lemma 5.6 either LEG_{H}(ϕ^{N}(α)_{H}) ≥ ϵ or LEG_{H}(ϕ^{−N}(α)_{H′}) ≥ ϵ. For concreteness, assume the former holds. If N ≥ N_{1}, we have

\begin{matrix}||ϕ^{N}(α)|| ≥ \frac{1}{K} length_{H}(ϕ^{N}(α)) ≥ \frac{1}{K} 3DK^{2}length_{H}(ϕ^{N_{0}}(α)) \\ ≥ \frac{1}{K} 3DK^{2} \frac{1}{K} ||ϕ^{N_{0}}(α)|| ≥ \frac{1}{K} 3DK^{2} \frac{1}{K} \frac{1}{D} ||α|| = 3||α||. \end{matrix}

(2) Denote by M the maximum length of ϕ̃^{s}(generator), s = 0, ±1, ±2, ⋯, ±N_{1}. We first argue that if |w| ≥ M − 3, then |ϕ̃^{N_{1}}(w)| ≥ 2|w| or |ϕ̃^{−N_{1}}(w)| ≥ 2|w|. Indeed, choose a generator x such that xw is cyclically reduced. Then by (1) we have |ϕ̃^{N_{1}}(xw)| ≥ 3|xw| or |ϕ̃^{−N_{1}}(xw)| ≥ 3|xw|. Say for concreteness that |ϕ̃^{N_{1}}(xw)| ≥ 3|xw|. Then |ϕ̃^{N_{1}}(w)| ≥ 3|xw| − |ϕ̃^{N_{1}}(x)| ≥ 3 + 3|w| − M ≥ 2|w|.

It now follows by a simple induction that for |w| ≥ M − 3 and any integer k either ϕ̃^{kN_{1}}(w) ≥ 2^{k}|w| or ϕ̃^{−kN_{1}}(w) ≥ 2^{k}|w|. Fix an integer k_{0} such that 2^{k_{0}} ≥ 2M. Any integer N ≥ k_{0}N_{1} can be written as N = kN_{1} + l with k ≥ k_{0} and 0 ≤ l < N_{1}. We now see that for all such N we must have |ϕ̃^{N}(w)| ≥ 2|w| or |ϕ̃^{−N}(w)| ≥ 2|w|. Indeed, if we assume for concreteness that |ϕ̃^{kN_{1}}(w)| ≥ 2^{k}|w|, then |ϕ̃^{N}(w)| = |ϕ̃^{kN_{1}+l}(w)| ≥ 2^{k}|w|/M ≥ 2|w|.

For every w with |w| < M −3 there is N_{w} such that |ϕ̃^{N}(w)| ≥ 2|w| for N ≥ N_{w}. It now suffices to take N′_{1} larger than all such N_{w}’s and larger than k_{0}N_{1}.

(3) This is now a consequence of the Combination Theorem in [BF2-3], since (2) implies the Annuli Flare Condition. □

Proof of Theorem 5.2. (1) Suppose that such N_{2} does not exist Let {β_{i}} be a sequence of conjugacy classes such that at least two of the lengths ||ϕ^{n_{i}}(β_{i})||, ||ϕ^{−n_{i}}(β_{i})||, ||ψ^{n_{i}}(β_{i})||, ||ψ^{−n_{i}}(β_{i})|| are < 3||β_{i}|| and n_{i} > i^{2}. By Theorem 5.1(1), for large i, possibly after replacing ϕ and/or ψ by their inverses, we may assume that ||ϕ^{n_{i}}(β_{i})|| < 3||β_{i}|| and ||ψ^{n_{i}}(β_{i})|| < 3||β_{i}||. Since ϕ and ψ don’t have common powers, Λ_{ϕ}^{+} ≠ Λ_{ψ}^{+}, and the sequence {β_{i}} cannot weakly converge to both Λ_{ϕ}^{+} and Λ_{ψ}^{+}. For concreteness, we assume that it does not converge to Λ_{ϕ}^{+}. By Corollary 5.7 it follows that, after perhaps passing to a subsequence, the sequence {ϕ^{−2i}(β_{i})} weakly converges to Λ_{ϕ}^{−}. In particular, for large i, we have LEG_{H′}(ϕ^{−2i}(β_{i})_{H′}) ≥ ϵ > 0. By Lemma 5.5(1) it follows that for some N_{3} > 0 and large i

length_{H′}(ϕ^{−n_{i}}(β_{i})_{H′}) ≥ (const)3^{\frac{n_{i}−2i}{N_{3}}} length_{H′}(ϕ^{−2i}(β_{i})_{H′}).

Since length_{H′}(ϕ^{−2i}(β_{i})) ≥ δ^{2i}length_{H′}(β_{i}) for some δ > 0 and since n_{i} > i^{2}, we conclude that length_{H′}(ϕ^{−n_{i}}(β_{i}))/length_{H′}(β_{i}) → ∞, contradicting the choice of the sequence {β_{i}}.

(2) Similar to the proof of Theorem 5.1(2).

(3) This is a consequence of (2) and the Combination Theorem of [BF2-3]. □

Proof of Theorem 5.3. When ϕ is geometric, the statement is well-known and follows from surface theory. See [LS]. Now suppose that ϕ is nongeometric. From Lemma 5.5 and Lemma 5.4(2) it follows that there is N and δ > 0 such that for every nontrivial α either l_{T^{−}}(ϕ^{N}(α)) ≥ δ or l_{T^{+}}(ϕ^{−N}(α)) ≥ δ. It follows that either l_{T^{−}}(α) ≥ \frac{δ}{λ^{N}} or l_{T^{+}}(α) ≥ \frac{δ}{λ^{′N}} where λ′ is the growth rate of ϕ^{−1}. □

Proof of Theorem 5.4. Again, the statement follows from surface theory when ϕ is geometric. Assume ϕ is nongeometric. Assume on the contrary that there is a sequence α_{i} ∈ A such that l_{T^{+}}(α_{i}) → 0. Thus LEG_{H′}(α_{i}) → 0 as well, and furthermore for any N > 0 LEG_{H′}(ϕ^{−N}(α_{i})) → 0. By Corollary 5.7 we conclude that the α_{i}’s contain longer and longer leaf segments of Λ^{+}, and hence Λ^{+} is carried by A, violating Proposition 2.4. □

Question: Let ϕ be irreducible. Is every tree in X other than T_{ϕ}^{−} attracted to T_{ϕ}^{+} under iteration by ϕ?

Appendix: The Tits Alternative for Out(F_{n}) – An outline

A group is said to satisfy the “Tits Alternative” if each subgroup either is virtually solvable or contains F_{2}. In this appendix, we give an outline of our proof of the following.

Main Theorem. The group Out(F_{n}) satisfies the Tits Alternative. Further, a solvable subgroup of Out(F_{n}) has a subgroup of index at most 3^{5n^{2}} that is finitely generated and free abelian.

The rank of an abelian subgroup of Out(F_{n}) is bounded by vcd(Out(F_{n})) = 2n − 3 for n > 1 [CV].

The proof is contained in a series of 3 papers [BFH1], [BFH2], and [BFH3]. The first paper contains a reduction to the case where H consists entirely of polynomially growing elements; such groups are called P G. In the second paper, it is shown that P G subgroups are either virtually solvable or contain F_{2}. Finally in the third paper, solvable subgroups of Out(F_{n}) are proved to be virtually abelian.

The reduction to P G subgroups is similar in spirit to the proofs of N. Ivanov [I] and J. McCarthy [McC] that the mapping class groups of compact surfaces satisfy the Tits Alternative. Let H be a subgroup of Out(F_{n}). There is associated to O a pair of laminations. For us a lamination is a closed subset of the space B_{n} := ((∂F_{n} × ∂F_{n}) \ Δ)/(F_{n} × ℤ/2ℤ) of bi-infinite paths in F_{n} where Δ is the diagonal, F_{n} acts diagonally, and ℤ/2ℤ acts by interchanging the factors.

More generally, if O is not necessarily irreducible, then to each exponentially growing stratum of a relative train track for O there corresponds a pair of laminations. The resulting finite set L(O) of laminations associated with O is independent of the particular relative train track representative. As in the irreducible case, if Λ ∈ L(O) and if H virtually stabilizes Λ then there is a “stretch” map H → ℤ. Also, if H virtually stabilizes neither Λ nor the lamination paired with Λ then a familiar ping-pong argument (see e.g. [Ti], [I], or [McC]) applied in the space B shows that H contains F_{2}. In contrast to the irreducible case, the kernel of the stretch can contain F_{2}.

If H does not contain F_{2} then it follows that if U ∈ H and if Λ ∈ L(U) then H virtually stabilizes either Λ or the lamination paired with Λ. It is shown that there are finitely many laminations virtually stabilized by H such that the product of their stretch maps has a kernel that is P G. Thus, the question of whether subgroups of Out(F_{n}) are either virtually solvable or contain F_{2} is equivalent to the same question for P G subgroups of Out(F_{n}). We have our reduction.

Before leaving exponentially growing automorphisms behind, let’s look at a couple of examples. Let O ∈ Out(< a, b >) be given by the automorphism Ô(a) = ab and Ô(b) = bab. The irreducible outer automorphism O represents a mapping class of a punctured torus. It has a pair of invariant laminations Λ^{+} and Λ^{−} obtained by taking the limit sets in B_{2} of {O^{i}(a)|i ∈ ℕ} and {O^{−i}(a)|i ∈ ℕ} respectively. Let U ∈ Out(< a, b, c, d >) be given by the automorphism Û whose restriction to < a, b > is Ô and such that Û(c) = cd and Û(d) = dcad. A relative train track for U has two strata corresponding to < a, b > and < c, d > respectively. The laminations Λ^{±} are the pair corresponding to the first stratum. The pair Σ^{±} corresponding to < c, d > is obtained as above by iterating say c under U^{±1}. Notice that Σ^{+} is in the closure of Λ^{+} yet Σ^{+} ≠ Λ^{+}. This phenomenon does not occur for surface laminations. For the ping-pong arguments, it is important to have a characterization of the attracting basin of a lamination in B_{n}. This is provided by Theorem 7.1 of [BFH1]. For example, any element of B_{2} other that Λ^{−} and the axis of [a, b] is attracted to Λ^{+} under iteration by O. Every element of B_{4} other than Σ^{−} and an element of B_{2} ⊂ B_{4} is attracted to Σ^{+} under iteration by U. The general situation is more subtle since strata need not be exponentially growing. The main technical tool is an improved version of the relative train track results of [BH]. See Theorem 6.4 of [BFH1].

In [BFH2] we show that P G subgroups H_{0} of Out(F_{n}) are either virtually solvable or contain F_{2}. There is a marked difference with the surface case. A P G subgroup of a mapping class group of a surface is virtually a subgroup of a group generated by Dehn twists in pairwise disjoint simple closed curves. The P G subgroup of Out(< a, b, c >) consisting of elements of the form a ↦ a, b ↦ b, and c ↦ ucv where u and v are words in a and b is isomorphic to F_{2} × F_{2}. Here, results more closely parallel the linear case and the techniques are tree theoretic. We will describe an analogue of a Kolchin theorem that is the main result of [BFH2].

Recall that a linear map is unipotent if it has a basis with respect to which it is upper triangular with 1’s on the diagonal. A celebrated theorem of Kolchin [Se] states that for any group of unipotent linear maps there is a basis with respect to which all elements of the group are upper triangular with 1’s on the diagonal.

An outer automorphism is unipotent if it is both polynomially growing and has a unipotent action on H_{1}(F_{n}; ℤ). A subgroup of Out(F_{n}) is UP G if all of its elements are unipotent. A P G subgroup is virtually UP G. As our Kolchin analogue, we show that for every finitely generated UP G subgroup H_{0} there is a nontrivial simplicial F_{n}-tree T with all edge stabilizers trivial that is fixed by all elements of H_{0}. This is Theorem 5.1 of [BFH2] and is our main result (together with some refinements) on the structure of UP G subgroups. The Tits Alternative for P G subgroups follows without much more work.

The tree T in the previous paragraph is found using a dynamical argument. If O ∈ Out(F_{n}) is unipotent and if T_{0} is a tree in the closure X of Culler-Vogtmann’s Outer Space CV then T_{0}O^{∞} := lim_{i→∞}T O^{i} exists and is simplicial if T_{0} is simplicial. Suppose that H_{0} =< O_{1}, ⋯, O_{k} > is a UP G subgroup. Suppose that T_{0} is a nontrivial simplicial tree with trivial edge stabilizers such that the set of elliptic elements of T_{0} is H_{0}-invariant and maximal with respect to this property. Consider the sequence T_{i+1} := T_{i}O_{i+1}^{∞} where the subscripts of the O’s are taken modulo k. This is called a “bouncing sequence” and it is shown to be eventually constant, the stable value being the desired tree T. See Theorem 5.5 of [BFH2].

We are now reduced to following. Let H be a subgroup of Out(F_{n}) that does not contain F_{2}. Then there is a product of a stretch maps with image a finitely generated free abelian group and P G kernel H_{0}. It remains to examine this extension. This is done in [BFH3]. First the Kolchin analogue is used to show that UP G subgroups are abelian. Thus, after perhaps passing to a subgroup of finite index, the kernel in our extension is abelian. It was known to Bass-Lubotzky [BL] that solvable subgroups of Out(F_{n}) are virtually polycyclic and in particular finitely generated. Thus, our kernel is a finitely generated torsion-free abelian group. (We give an independent proof.)

Next, we show that there are only finitely many combinatorial types in S := {SO^{∞}|S ∈ CV, O ∈ H_{0}}. (Two trees have the same combinatorial type if they are equivariantly homeomorphic.) Since S is H-invariant there is a combinatorial type and hence a simplicial tree T ∈ X that is virtually fixed by H.

A structure theorem for a group of automorphisms fixing a very small real F_{n}- tree is next developed, cf. [BJ]. Since this result is for subgroups of automorphisms rather than for outer automorphisms, it is important to be able to lift certain groups from Out(F_{n}) to Aut(F_{n}). The techniques of [BFH1] are used to obtain the following, cf. Proposition 10.8 of [BFH3].

Proposition. Let H be a solvable subgroup of Out(F_{n}) and let γ ∈ F_{n} be Hperiodic. Then H virtually lifts to Aut(F_{n}) fixing γ.

Finally, the structure theorem is used to further analyze our extension. We are able to conclude that H is virtually abelian.

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University of Utah, Salt Lake City, UT 84112 E-mail address: bestvina@math.utah.edu

Rutgers University, Newark, NJ 07102 E-mail address: feighn@andromeda.rutgers.edu

CUNY, Lehman College, Bronx, NY 10468 E-mail address: chflc@cunyvm.cuny.edu