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004 Parageometric outer automorphisms of free groups

Michael Handel and Lee Mosher

February 6, 2019

Abstract

We study those completely irreducible outer automorphisms Φ of a finite rankfree group Fr which are parageometric, meaning that the attracting fixed pointof Φ in the boundary of outer space is a geometric R-tree with respect to theaction of Fr, but Φ itself is not a geometric outer automorphism in that it isnot represented by a homemorphism of a surface. Our main result shows thatthe expansion factor of Φ is strictly larger than the expansion factor of Φ−1. Ascorollaries (proved independently by Guirardel), the inverse of a parageometricouter automorphism is neither geometric nor parageometric, and a completelyirreducible outer automorphism Φ is geometric if and only if its attracting andrepelling fixed points in the boundary of outer space are geometric R-trees.

1 Introduction

There is a growing dictionary of analogies between theorems about the mappingclass group of a surface MCG(S) and theorems about the outer automorphism groupof a free group Out(Fr). For example, the Tits alternative for MCG(S) [McC85]is proved using Thurston’s theory of measured geodesic laminations [FLP+79], andfor Out(Fr) it is proved using the Bestvina–Feighn–Handel theory of laminations([BFH97], [BFH00], [BFH04]).

Expansion factors. Here is a result about MCG(S) which one might hope to havean analogue in Out(Fr). Given a finitely generated group G, its outer automorphismgroup Out(G) acts on the set of conjugacy classes C of G. Given c ∈ C let ‖c‖ be thesmallest word length of a representative of c. Given Φ ∈ Out(G) define the expansionfactor

λ(Φ) = supc∈C

(lim supn→+∞

‖Φn(c)‖1/n)

If Φ ∈ MCG(S) ≈ Out(π1S) is pseudo-Anosov then λ(Φ) equals the pseudo-Anosovexpansion factor [FLP+79]. By combining results of Thurston and Bers one obtains:

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Theorem. If Φ ∈ MCG(S) is pseudo-Anosov, then there is a unique Φ-invariantgeodesic in Teichmuller space, consisting of the points in Teichmuller space whichminimize the translation distance under Φ. This translation distance equals log(λ(Φ)).

Consider a free group Fr and the Culler-Vogtmann outer space Xr on whichOut(Fr) acts properly. Recall that Φ ∈ Out(Fr) is reducible if there is a nontrivialfree decomposition Fr = A1 ∗ · · · ∗Ak ∗B such that Φ permutes the conjugacy classesof A1, . . . , Ak; otherwise, Φ is irreducible. If Φk is irreducible for all k ≥ 1 then wesay that Φ is completely irreducible. By analogy, a mapping class on S is reducible ifit preserves the isotopy classes of some nontrivial decomposition of S into essentialsubsurfaces, and is completely irreducible if and only if it is represented by a pseudo-Anosov homeomorphism.

Question: Does the above theorem have an analogue for a completely irreducibleΦ ∈ Out(Fr)? The question does not quite make sense because no metric is specifiedon Xr, but one can instead ask: Is there an analogue with respect to some Out(Fr)-equivariant metric on Xr? Or on any other metric space on which Out(Fr) acts?

Answer: No. If translation distance for Φ is uniquely minimized on an axis γ then,by symmetry of the distance function, translation distance for Φ−1 is also uniquelyminimized on γ, and the minima for Φ and for Φ−1 along γ are equal. By equatingminimal translation distance with log(λ), one would conclude that λ(Φ) = λ(Φ−1).However, an example from [BH92] has the property that λ(Φ) 6= λ(Φ−1): considerΦ ∈ Out(F3) and Φ−1 represented by the automorphisms

φ :

A → AC

B → A

C → B

φ−1 :

A → B

B → C

C → BA

Interpreting these formulas as self maps of the three-petaled rose, each is clearly atrain track map. From the results of [BH92] (see also Proposition 4), if f : G → Gis a train track representative of Φ ∈ Out(Fr), and if PF(Mf ) denotes the Perron-Frobenius eigenvalue of the transition matrix Mf , then λ(Φ) = PF(Mf ). For theabove two train track maps we therefore obtain

λ(Φ) = PF

1 0 11 0 00 1 0

> 1.4, λ(Φ−1) = PF

0 1 00 0 11 1 0

< 1.4

Confronted with such a strange phenomenon, one strategy is to see what appro-priately weaker results can be proved. We follow this strategy in the companionpaper [HM04a] where we show that the ratio log(λ(Φ))/ log(λ(Φ−1)) is bounded bya constant depending only on the rank r. This is what one would expect if there

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were an axis γ for Φ with translation distance log(λ(Φ)) and an axis γ′ for Φ−1 withtranslation distance log(λ(Φ−1)), such that γ, γ′ are fellow travelers. Encouraged bythis result, we have pursued the study of axes in outer space, with some interestinganalogues of uniqueness of axes [HM04b].

Parageometric outer automorphisms. In this work we pursue another strategy:explore the strange phenomenon on its own terms. The Φ described above turns out tobe an example of a parageometric outer automorphism, as we discovered by examiningdiscussions of this same example in [BF95] and in [BF]. The interest in this conceptwas pointed out in [GJLL98], where we found the terminology “parageometric”.

While we believe that the phenomenon λ(Φ) 6= λ(Φ−1) is probably generic, weshall show that this inequality is always true when Φ is parageometric, in which casewe give an explicit geometric argument which shows that λ(Φ) > λ(Φ−1).

To define parageometricity, recall the action of Out(Fr) on the compactified outerspace X r = Xr ∪ ∂Xr consisting of (classes of) very small actions of Fr on R-trees(see [BF] and also [CL95]); we shall call these objects “Fr-trees”. The action of acompletely irreducible Φ ∈ Out(Fr) on X r has source–sink dynamics, with a repellingFr-tree T− ∈ ∂Xr, and an attracting Fr-tree T+ ∈ ∂Xr (see [LL03] and also [BFH97]).An Fr-tree is geometric if it is dual in the appropriate sense to a measured foliationdefined on some 2-complex whose fundamental group surjects to Fr [LP97]. Forexample, if a completely irreducible Φ ∈ Out(Fr) is geometric, meaning that it isrepresented by an automorphism of a surface with boundary, then both of the Fr-trees T−, T+ are geometric: this follows from Thurston’s theorem that Φ is representedby a pseudo-Anosov surface homeomorphism f : S → S, because T− and T+ are dualto the stable and unstable measured foliations of f defined on the surface S.

A completely irreducible Φ ∈ Out(Fr) is said to be parageometric if T+ is ageometric Fr-tree but Φ is not a geometric outer automorphism.

For example, the outer automorphism Φ described above is parageometric: geo-metricity of the Fr-tree T+ is proved in Example 3.4 of [BF]; and Levitt’s “thinness”property for T− is proved in Example 10.1 of [BF95], showing that T− is not a ge-ometric Fr-tree, and so Φ is not a geometric outer automorphism. In Proposition 5we will gather results of [BF] and [BH92] which give a method of characterizingparageometricity solely from the properties of a train track representative.

Here are our main results:

Theorem 1. If Φ ∈ Out(Fr) is parageometric then λ(Φ) > λ(Φ−1).

Corollary 2. If Φ ∈ Out(Fr) is parageometric then Φ−1 is neither geometric norparageometric.

Proof. If Φ−1 is geometric then λ(Φ−1) = λ(Φ), whereas if Φ−1 is parageometricλ(Φ−1) > λ((Φ−1)−1) = λ(Φ). ♦

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Corollary 3. A completely irreducible Φ ∈ Out(Fr) is geometric if and only if theFr-trees T− and T+ are both geometric.

Proof. If Φ is not geometric then Φ−1 is also not geometric, but if T−, T+ wereboth geometric trees then it would follow by definition that Φ and Φ−1 are bothparageometric, contradicting Corollary 2. The other direction was noted above. ♦

Corollaries 2 and 3 have been proved independently by Guirardel [Gui04], bydifferent means.

Theorem 1 and Corollary 2 were first presented at various seminars in the Fallof 2003, including the Topology Seminar at Princeton University. Our thanks go toBaris Coskunuzer of that seminar for a question which quickly inspired the proof ofthe “if” direction of Corollary 3.

Sketch of the proof of Theorem 1. In Section 2 we recall results from [BF] thatcharacterize when T+ is a geometric Fr-tree. This happens if and only if (some powerof) Φ has a train track representative g : G→ G for which there is a unique periodicNielsen path ρ of length 2L = 2L(G). This Nielsen path is necessarily fixed, and itdecomposes as ρ = α ∗ β where α, β are legal paths of length L and the turn at theconcatenation point is the unique illegal turn of the train track representative g. Inthis situation, following [BF] we construct a 2-dimensional dynamical system k : K →K representing Φ, which we call the wedge model, by attaching to G a wedge W , atriangle with one side attached along α and the other side attached along β. Theunattached side of W is vertical, and each vertical segment of W has endpoints on acorresponding pair of points, one in α and one in β. The effect of g : G→ G is to foldρ by some amount, and this extends to k : K → K by collapsing vertical segments ofW to some amount. The vertical segments of W form leaf segments of a measuredfoliation F on K, the stable foliation of g, whose transverse measure restricts to theLebesgue measure along the train track G. Again following [BF], in this situation weshow that the attracting tree T+ is the dual tree of the measured foliation F .

Combining this construction with results of [BH92], we show that Φ is parageo-metric if and only if the two endpoints of ρ are distinct, which happens if and onlyif there exists an edge of G which is covered exactly once by the Nielsen path ρ, andso is a free edge of K.

The main theorem is proved by computing the asymptotic compression factor ofk on leaves of F in two different ways. One interpretation of this factor is as theexponential growth rate in n for the number of vertical segments in a leaf of F thatkn can collapse to a point. This rate is simply the Perron-Frobenius eigenvalue of thetransition matrix for g : G→ G, which equals λ(Φ). But we are particularly interestedin the collection of bi-infinite lines contained in leaves of F , which we denote H(F),

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the hull of F . Because some edge of G is a free edge of K, the interior of that edge isdisjoint from any line in the hull of F , and so the asymptotic compression factor of krestricted to lines in H(F) is strictly less than the Perron-Frobenius eigenvalue λ(Φ).This is the culminating argument of the proof, contained at the end of Section 4. Onthe other hand, we use properties of k : K → K to essentially identify H(F) withthe expanding lamination of Φ−1, allowing us to equate its asymptotic compressionfactor with λ(Φ−1). This argument is carried out in Section 3.

When does λ(Φ) = λ(Φ−1)? Consider a completely irreducible Φ ∈ Out(Fr). Inthe wake of our results one might wonder whether λ(Φ) = λ(Φ−1) implies that Φ isgeometric. Here is an easy construction of counterexamples.

In any group G, if g, g′ ∈ G have order 2 then gg′ and (gg′)−1 are conjugate. Itfollows that if Ψ,Ψ′ ∈ Out(Fr) have order 2, and if Φ = ΨΨ′ is completely irreducible,then λ(Φ) = λ(Φ−1).

For a concrete example, let Ψ ∈ Out(F3) be the order two element represented bythe automorphism a 7→ b, b 7→ a, c 7→ c. Let Ψ′ = ΘΨΘ−1 be a conjugate of Ψ. If theconjugating element Θ is picked randomly then one might expect that Φ = ΨΨ′ iscompletely irreducible and has a train track representative with no periodic Nielsenpaths, and so Φ is nongeometric. Taking Θ to be the fourth power of the outerautomorphism a 7→ b, b 7→ c, c 7→ ba considered earlier, and applying the train trackalgorithm of [BH92], one obtains a train track map g : G→ G as follows. The graphG has two vertices r, q, four edges B,C,D,E, with B from q to r, C from r to q, Dfrom r to q, and E from q to q, and g is defined by g(B) = CECDE, g(C) = CBED,g(D) = B, and g(E) = CB. The expansion factor is λ = 3.199158087 . . ..

To see that g represents a nongeometric, completely irreducible outer automor-phism requires three things to be checked. First, g is a completely irreducible traintrack map. Second, at each of the two vertices v = r, q of G, the graph of turns takenat v is connected; this is the graph with one vertex for each direction at v and oneedge for each turn at v that is taken by the g image of some edge of G. Third, ghas no Nielsen paths, which can be checked by the following expedient. Factor g intoStallings folds, G = G0 → G1 → · · · → Gn = G; we did this with n = 8. Let Γ0 bethe set of length 2 edge paths in G0 with an illegal turn; there are two such pathsup to reversal, ED and EB. For negative integers i define Γi inductively to be aset of edge paths in Gi (with indices taken modulo n) each with one illegal turn, asfollows: for each γ ∈ Γi, take all paths in Gi−1 with one illegal turn whose straight-ened image in Gi is γ, and put each such path in Γi−1. Carrying this process out, wecomputed that the set Γ−12 is empty. This shows that g has no Nielsen paths, andso it represents a completely irreducible, nongeometric outer automorphism.

Just as a check, we also inverted the sequence of Stallings folds and applied the

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train track algorithm to verify that the expansion factor of the inverse is also equalto 3.199158087 . . ..

To get wider classes of examples, consider a completely irreducible Φ ∈ Out(Fr)with expanding lamination denoted Λu. By Section 2 of [BFH97] the group Out(Fr)acts on the set of expanding laminations of completely irreducible elements, and thereis a homomorphism ℓu : Stab(Λu) → R+ with discrete image and finite kernel suchthat ℓu(Φ) = λ(Φ), and ℓu(Ψ) = λ(Ψ) as long as ℓu(Ψ) ≥ 1. Applying this to Φ−1 withexpanding lamination denoted Λs, we obtain a homomorphism ℓs : Stab(Λs) → R+

such that ℓs(Φ−1) = λ(Φ−1) and ℓs(Ψ) = λ(Ψ) as long as ℓs(Ψ) ≥ 1. Applying Propo-sition 2.16 of [BFH00] it follows that Stab(Λs) = Stab(Λu), a subgroup of Out(Fn)that we denote VΦ (note that VΦ is the virtual centralizer of 〈Φ〉 in Out(Fr), consist-ing of all Ψ ∈ Out(Fr) that commute with some positive power of Φ; VΦ is containedin the virtual centralizer because 〈Φ〉 has finite index in VΦ; and if Ψ 6∈ VΦ thenΨΦkΨ−1 6= Φk because their attracting fixed points in ∂Xr are distinct, by Proposi-tion 2.16 of [BFH00]). From the properties of the homomorphisms ℓu, ℓs : VΦ → R+

it follows that the infinite cyclic group 〈Φ〉 has finite index in VΦ, and so any twoelements of VΦ not contained in the common kernel of ℓs, ℓu have nonzero powers thatare equal. This implies that if λ(Φ) = λ(Φ−1) then λ(Ψ) = λ(Ψ−1) for any Ψ ∈ VΦ.

It might be interesting to find necessary and sufficient conditions for the conditionλ(Φ) = λ(Φ−1), for completely irreducible Φ ∈ Out(Fr). For example, is it necessarythat VΦ contains a completely irreducible element that is either geometric or conjugateto its own inverse? At the very least, it would seem that the property λ(Φ) 6= λ(Φ−1)is generic, and the property that Φ be completely irreducible with nongeometric fixedtrees T−, T+ is also generic.

2 Preliminaries

2.1 Outer automorphisms and outer space.

The definitions in this section follow several sources. For the foundations of markedgraphs, R-trees, and outer space, including many of the facts recalled below withoutcitation, see [CM87] and [CV86]. For concepts of irreducibility see [BH92]. A goodoverview is given in [Vog02].

Outer automorphisms of free groups. Fix an integer r ≥ 2, let Fr denote thefree group of rank r, let Out(Fr) = Aut(Fr)/ Inn(Fr) denote its outer automorphismgroup, and let C denote its set of nontrivial conjugacy classes. Let Rr denote the rosewith r-petals and identify π1(Rr) ≈ Fr, so the group Out(Fr) is identified with thegroup of homotopy classes of self-homotopy equivalences of Rr. Given Φ ∈ Out(Fr)let fΦ : Rr → Rr be a representative homotopy equivalence. Out(Fr) acts naturally

6

on C, and on conjugacy classes of subgroups of Fr. We say that Φ ∈ Out(Fr) isreducible if there exists a nontrivial free factorization Fr = A1 ∗ · · · ∗ Ak ∗B so thatΦ permutes the conjugacy classes of A1, . . . , Ak. If Φ is not reducible then it isirreducible. If Φn is irreducible for all n ≥ 1 then Φ is completely irreducible1. Notethat Φ is irreducible if and only if Φ−1 is, and the same for complete irreducibility.

Outer space and its boundary. An Fn-tree is an R-tree T equipped with anaction of Fn that is minimal (no proper nonempty subtree is invariant) and nonele-mentary (T is not a point or a line). An Fr-tree is proper if the action is properlydiscontinuous, and it is simplicial if T is a simplicial complex. Two Fn trees are iso-metrically (resp. homothetically) conjugate if there is an isometry (resp. homothety)between them that conjugates one action to the other. Outer space Xr is the set ofhomothetic conjugacy classes of proper, simplicial Fn-trees, with topology induced byembedding Xr → PRC as follows: first embed the set of isometric conjugacy classesinto RC using translation length as a class function on Fn, and then projectivize. Theimage of this embedding is precompact, and its closure and boundary are denotedX r and ∂Xr = X r − Xr.

Points of outer space can also be represented as marked graphs, as follows. Amarked graph is a graph G with all vertices of valence ≥ 3, equipped with a pathmetric, and with a homotopy equivalence Rr → G called the marking. If a base pointp ∈ G happens to be imposed, the homotopy class of a marking of G determines andis determined by an isomorphism Fr → π1(G, p) up to precomposition by an innerautomorphism of Fr. Two marked graphs G,G′ are isometric (resp. homothetic)if there exists an isometry (resp. homothety) G → G′ which, together with themarkings, makes the following diagram commute up to homotopy:

G((

Rroo // G′

Passage to the universal covering space induces a bijection between the set of homo-thety classes of marked graphs and the set Xr. The embedding Xr → PRC can beunderstood by first associating to a marked graph G the class function on Fn thatassociates to an element of Fn the length of the shortest loop in G representing thefree homotopy class of that element, and then projectivizing.

The group Out(Fr) acts on X r on the right, as follows. Let [·] denote homothetyclass. For each Fr-tree T one can precompose the action Fr → Isom(T ) with anautomorphism Fr → Fr representing Φ, to get [T ]Φ. In terms of a marked graph G,one can precompose the marking Rr → G with a homotopy equivalence Rr → Rr thatrepresents Φ, to obtain [G]Φ. This action preserves the topology, and it preservesouter space Xr itself and its boundary ∂Xr.

1called “irreducible with irreducible powers” or “IWIP” in the literature

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Source–sink dynamics. If Φ ∈ Out(Fr) is completely irreducible then there existT− 6= T+ ∈ ∂Xr such that for every x ∈ X r,

limn→−∞

Φn(x) = T− and limn→+∞

Φn(x) = T+

where these limits take place in X r. This was proved for x ∈ Xr in [BFH97] andextended to all x ∈ X r in [LL03]. We call T− the repelling tree and T+ the attractingtree of Φ.

Geometric Fr-trees. We review measured foliations on 2-complexes and geometrictrees following [LP97].

Consider a connected simplicial 2-complex K which is not a point. A measuredfoliation on K is defined by specifying measured foliations on each 2-simplex of Kwhich fit together compatibly along 1-simplices. To be precise, a measured folia-tion on a 1 or 2 dimensional simplex σ in K is determined by choosing a simplicialhomeomorphism σ′ → σ where σ′ is a rectilinear simplex in R2, and pushing for-ward the vertical foliation on R2 with the transverse measure |dx|. A leaf segmentin σ is the pushforward of σ′ intersected with a vertical line; for example, if σ is a1-simplex then either σ is a single leaf segment or each point of σ is a leaf segment.A measured foliation on a 2-simplex restricts to a measured foliation on each of itsedges. A measured foliation on K is determined by choosing a measured foliation oneach 2-simplex of K, so that for each 1-simplex e, all of the measured foliations on eobtained by restricting to e from a 2-simplex incident to e agree with each other.

We will often suppress the simplicial structure on K, so a measured foliation ona cell complex means, formally, a measured foliation on some simplicial subdivision.

Let F denote a measured foliation on K, and F∣∣ σ its restriction to each simplex

σ of K. The collection of leaf segments in 1-simplices and 2-simplices define a relationon K, two points being related if they are contained in the same leaf segment. Thisrelation generates an equivalence relation on K. The equivalence classes are calledleaves. The leaf containing a point x ∈ K can be built up inductively as follows: letℓ1 be the union of all leaf segments containing x; for i ≥ 1 let ℓi+1 be the union of allleaf segments containing points of ℓi; finally, the leaf containing x is ∪∞

i=1ℓi.Given a measured foliation F on K and a path γ : I → K, pulling back the

tranverse measure locally gives a measure on I, whose integral is denoted∫γ F .

Given a finite 2-complex K with measured foliation F and a surjective homomor-phism h : π1(K) → Fr, let K → K be the covering space corresponding to ker(h),and let F be the lifted measured foliation on K. Define a pseudo-metric on K whered(x, y) is the infimum of the transverse measures of paths from x to y. Let T be theassociated metric space, whose points are the equivalence classes determined by therelation d(x, y) = 0. Note that if x, y are in the same leaf of K then d(x, y) = 0, but

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the converse need not hold in general. The action of Fr on K induces an isometricaction of Fr on the metric space T . We assume that each 1-simplex e of K is ageodesic, that is, if ∂e = {v,w} then d(v,w) =

∫e F . Under this assumption, Levitt

and Paulin [LP97] prove that T is an Fr-tree, called the dual Fr-tree of the measuredfoliation F .

An Fr-tree T is said to be geometric if there exists a finite 2-complex K withmeasured foliation F , and a surjective homomorphism π1(K) ≈ Fr, such that eachedge of K is a geodesic, and such that T is isometrically conjugate to the dual Fr-treeof F .

Geometric and parageometric outer automorphisms. An outer automor-phism Φ ∈ Out(Fr) is geometric if there exists a compact surface S, an isomorphismFr ≈ π1S, and a homeomorphism h : S → S, such that the outer automorphism ofFr induced by h is equal to Φ. If Φ is completely irreducible and geometric, then itsattracting and repelling Fr-trees T+, T− are both geometric Fr-trees.

Consider now a completely irreducible Φ ∈ Out(Fr) with attracting Fr-tree T+ ∈∂Xr. We say that Φ is parageometric if T+ is a geometric Fr-tree but Φ is not ageometric outer automorphism.

2.2 Train tracks and laminations

The definitions in this section follow [BH92] and [BFH97].

Topological representatives and Markov partitions. Given Φ ∈ Out(Fr), amarked graph G with marking µ : Rr → G, and a homotopy equivalence g : G → G,we say that g is a topological representative of Φ if g takes vertices to vertices, g isan immersion on each edge, and the following diagram commutes up to homotopy:

RrfΦ //

µ

��

Rr

µ

��

G g// G

and so the composition Rr → Gg−→ G represents the point [G]Φ ∈ Xr. The set of

edges E of G forms a Markov partition for g, meaning that for any e, e′ ∈ E , eachcomponent of e ∩ g−1(int(e′)) is mapped by g homeomorphically onto int(e′). Thetransition graph Γ of g is a directed graph whose vertex set is the set E , such thatfor each e, e′ ∈ E , the directed edges from e to e′ are in one-to-one correspondencewith the components of e ∩ g−1(int(e′)). The transition matrix of g is the functionM : E ×E → Z, whereM(e, e′) equals the number of directed edges in Γ from e to e′,

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in other words, the number of times that g(e′) traverses e in either direction. Notethat Mn(e, e′) is the number of directed paths from e to e′ of length n. We say thatΓ, M , and g : G→ G are irreducible if if there is an oriented path from any vertex ofΓ to any other vertex, equivalently, for all e, e′ there exists n such that Mn(e, e′) 6= 0.If irreducibility holds then the Perron-Frobenius theorem implies that there exists aunique λ ≥ 1 such that M has a positive (right) eigenvector with eigenvalue λ. IfM has the stronger property that some positive power has all positive entries thenλ > 1 and a positive eigenvector is unique up to positive scalar multiple.

Train tracks. A direction of G at a vertex v is the germ, up to reparameterization,of an immersed path with initial point v. Each direction is uniquely represented byan oriented edge e with initial point v, but we occasionally use other paths withinitial point v to represent directions. A turn of G at v is an unordered pair {e, e′} ofdirections at v; the turn is nondegenerate if e 6= e′, otherwise the turn is degenerate.

An edge path in G will always mean a concatenation of the form γ = e0 ∗ e1 ∗ · · · ∗ek−1 ∗ ek, k ≥ 0, where e1, . . . , ek−1 are oriented edges and e0, ek are subsegments oforiented edges. Often we say “path” when “edge path” is meant; the context shouldmake this clear. Given an edge path γ : I → G and t ∈ int(I) so that γ(t) is a vertexof G, let e, e′ be the two directions of γ at this point, that is: subdivide at t to obtaina concatenation γ = α ∗ β, let e be the direction of α at its initial point, and let e′

be the direction of β at its initial point. With this notation we say that γ takes theturn {e, e′} at the parameter value t. If t is understood then we just say that γ takesthe turn {e, e′}.

A topological representative g : G → G acts on the set of directions and on theset of turns of G. A nondegenerate turn is illegal if its image under some positivepower of g is degenerate, otherwise the turn is legal. An edge path α is legal if everyturn taken by α is legal, in particular every legal path is immersed. Given a path αin G, let α# denote the immersed path (or constant path) which is homotopic to αrel endpoints, so if α is immersed then α# = α, and if α is legal then gn(α)# = gn(α)for all n ≥ 0.

A topological representative g : G→ G of Φ ∈ Out(Fr) is a train track representa-tive of Φ, and g is a train track map, if for each edge E of G, the map g

∣∣ E : E → Gis a legal path, equivalently, gn

∣∣ E is an immersion for each n ≥ 1. A train trackmap g is completely irreducible if gn is irreducible for all n ≥ 1.

Proposition 4 ([BH92]). If Φ ∈ Out(Fr) is completely irreducible then Φ has acompletely irreducible train track representative g : G → G. The transition matrixMg has a positive power, and so there exists λ(g) > 1 and vector v : E → R, withλ(g) unique and v unique up to a positive scalar multiple, so that Mv = λ(g)v. Wealso have λ(Φ) = λ(g) (see Remark 1.8 of [BH92]).

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With g : G → G as in this proposition, we may assign a path metric to G, alsocalled the Lebesgue measure on G, so that each edge e has length LG(e) = v(e). Wemay then alter g on each edge e by a homotopy rel endpoints so that g

∣∣ e stretchespath length by a constant factor of λ(g); the resulting map is still a train trackmap. The number λ(g) = λ(Φ) is called the stretch factor of g. The notation L(·)is generally used to denote length, with an optional subscript to specify the space inwhich length is measured, e.g. LG(·).

Henceforth we always assume without comment that if g : G → G is a traintrack representative of a completely irreducible Φ ∈ Out(Fr) then g is completelyirreducible and g stretches path length on G by the constant factor λ(Φ).

The geodesic lamination of a free group. In this heading and the next wereview the results from [BFH97] concerning the construction and properties of theexpanding or unstable lamination Λu(Φ) of a completely irreducible Φ ∈ Out(Fr). Weuse here a slightly different point of view than in [BFH97], presenting laminations asHausdorff objects rather than non-Hausdorff.

Consider a marked graph G with universal cover T . The geodesic laminationof T , denoted ΛT , is the set of pairs (ℓ, x) where ℓ ⊂ T is a bi-infinite, unorientedline and x ∈ ℓ, equipped with the compact open topology where a neighborhoodUǫ of (ℓ, x) is the set of all (ℓ′, x′) such that d(x, x′) ≤ ǫ and ℓ ∩ ℓ′ contains a 1/ǫneighborhood of x in ℓ and a 1/ǫ neighborhood of x′ in ℓ′. The projection mapΛT → T is the map (ℓ, x) → x. A leaf of ΛT corresponding to a bi-infinite line ℓ isthe set of all (ℓ, x) such that x ∈ ℓ; we shall often confuse a bi-infinite line in T withits corresponding leaf. A point of ΛT can also be described as a geodesic embeddingγ : R → T modulo precomposition by the involution x↔ −x, where the point of ΛTcorresponding to γ is the pair (γ(R), γ(0)). The action of Fr on T induces a properlydiscontinuous, cocompact action on ΛT whose quotient space, a compact laminationdenoted ΛG, is the geodesic lamination of G. The projection map ΛT → T descendsto a projection map ΛG → G. An element of ΛG can also be described as a locallygeodesic immersion R → G modulo precomposition by the involution x↔ −x on theparameter domain R. A sublamination of any lamination is a closed subset that is aunion of leaves. By compactness of ΛG, every sublamination of ΛG is compact.

Given two marked graphs G,G′, any homotopy equivalence g : G→ G′ induces ahomeomorphism g∗ : ΛG → ΛG′ well defined up to isotopy, defined as follows. Firstalter g by homotopy so that it takes vertices to vertices and is affine on each edge. Liftg to the universal covers g : T = G→ G′ = T ′. There is an automorphism φ : Fr → Frsuch that g satisfies φ-twisted equivariance, meaning that f(g(x)) = g(φ(f)(x)) forall x ∈ T , f ∈ Fr. Consider a leaf ℓ of ΛT . Since g is a quasi-isometry, g(ℓ) is aquasi-geodesic embedding of R, and so the image g(ℓ) has finite Hausdorff distance

11

from some leaf that we shall denote g∗(ℓ). Define a function g# : ΛT → ΛT ′ that mapseach leaf ℓ to g∗(ℓ), by postcomposing the map g with the closest point projectionfrom g(ℓ) onto g∗(ℓ). The map g# is continuous, φ-twisted equivariant, and inducesa bijection of leaves. The image of the map ℓ→ g(ℓ) is the line g∗(ℓ) union a disjointset of finite trees attached to the line, and the effect of the closest point projectionis to collapse each of these finite trees to the point where it attaches to the line;it follows that the map g# is leafwise monotonic, meaning that for each leaf ℓ the

map ℓg#−−→ g∗(ℓ) has the property that each point pre-image is an arc. We can now

perturb g# to get a homeomorphism g∗, still satisfying φ-twisted equivariance, andg∗ is well-defined up to φ-twisted equivariant isotopy. It follows that g∗ descends tothe desired homeomorphism g∗, well-defined up to isotopy.

For any g : G → G′ as above that preserves the marking (in the sense that themarking Rr → G, postcomposed with g, is homotopic to the marking Rr → G′),note that the map g∗ : ΛG → ΛG′ is natural in the sense that for any two homotopy

equivalences Gg−→ G′ g′

−→ G′′ that preserve markings, the composition ΛGg∗−→ ΛG′

g′∗−→

ΛG′′ is isotopic to (g′ ◦ g)∗. We are therefore justified in talking about “the” geodesiclamination Λr of Fr, as represented by ΛG for any marked graph G. We are alsojustified in talking about a sublamination Λ′ ⊂ Λr of the geodesic lamination ofFr, represented as a sublamination Λ′

G ⊂ ΛG for any marked graph G, with theproperty that for any marking preserving homotopy equivalence g : G→ G′ we haveg∗(Λ

′G) = Λ′

G′ . We also say that Λ′G is the realization of Λ′ in the marked graph G.

Expanding laminations. Let Φ ∈ Out(Fr) be completely irreducible. The ex-panding or unstable lamination of Φ is a sublamination Λu(Φ) of the geodesic lam-ination of Fr, defined as follows. Choose any train track representative g : G → G.Choose an edge e and a periodic point x ∈ int(e) of periodicity p. As n → ∞, the

maps egnp

−−→ G can be reparameterized as a nested sequence of isometric immersionsof larger and larger subintervals of R, each interval containing 0 and each immersiontaking 0 to x. The union of these immersions is a bi-infinite geodesic in G, thatis, a leaf of ΛG. The closure of this leaf in ΛG is defined to be the realization ofΛu = Λu(Φ) in the marked graph G, denoted ΛuG. Λ

u is well-defined, independent ofthe choice of e and x, and also independent of g meaning that for any other traintrack representative g′ : G′ → G′ and any marking preserving homotopy equivalenceh : G→ G′ we have h∗(Λ

uG) = ΛuG′ .

The expanding lamination Λu(Φ) is minimal, meaning that its only nonemptysublamination is itself, in other words, every leaf is dense; see Section 1 of [BFH97].

Note that the action of g∗ on ΛG restricts to an action on ΛuG which expandslength by the exact factor of λ(Φ), that is, L(g∗(ℓ)) = λ(Φ)L(ℓ) for any leaf segmentℓ of ΛuG.

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2.3 Characterizing when T+ is geometric and Φ is parageometric

Consider a completely irreducible Φ ∈ Out(Fr) with attracting tree T+. In thissection we follow [BF] to characterize when T+ is a geometric tree, and combinedwith [BH92] we also characterize when Φ is parageometric. The characterizations areexpressed solely in terms of train track representatives of Φ or its powers. The prooffrom [BF] is included in detail, in part because the paper is not readily available, andalso because certain details are central to this paper, particularly the “wedge model”,a foliated 2-complex that exhibits geometricity of T+.

Consider a train track map g : G → G. A Nielsen path of g is a locally geodesicpath ρ : [0, a] → G such that ρ(0) and ρ(a) are fixed points, and g ◦ ρ is homotopicrel endpoints to ρ. A periodic Nielsen path of g is a Nielsen path of some power gn

with n ≥ 1. A (periodic) Nielsen path ρ is indivisible if it cannot be written as anontrivial concatenation of (periodic) Nielsen paths. Assuming that g is irreducible,every indivisible periodic Nielsen path has a unique decomposition ρ = α ∗ β whereα, β are legal paths of equal length, and ρ takes an illegal turn at the concatenationpoint. Depending on the context we may write this decomposition in other forms,for example, ρ = α1 ∗ α2. For details on Nielsen paths see [BH92].

In a 2-dimensional cell complex K, a free edge is an edge whose interior pointseach have a neighborhood homeomorphic to (−∞,+∞)× [0,∞).

Proposition 5 ([BF], [BH92]). Let Φ ∈ Out(Fr) be completely irreducible. LetT+ ∈ ∂Xr be the attracting tree of Φ. There exists k ≥ 1 and a train track represen-tative g : G→ G of Φk such that one of the following holds:

(1) g has no periodic Nielsen paths. This holds if and only if T+ is not geometric.

(2) g has a unique indivisible periodic Nielsen path ρ up to orientation reversal, ρis a fixed Nielsen path, the image of ρ is all of G, L(ρ) = 2L(G), and the illegalturn taken by ρ is the unique illegal turn of g. This happens if and only if T+is geometric.

Assume now that T+ is geometric, with ρ : [0, 2L(G)] → G as in (2). Let K be the2-complex obtained from the disjoint union of G with a closed disc W , identifyingpoints of an embedded arc γ : [0, 2L(G)] → ∂W with points of G via the identificationγ(t) ∼ ρ(t).

(3) Φ is geometric if and only if ρ is a closed path that traverses every edge ofG exactly twice. This occurs if and only if K is a compact surface with oneboundary component.

(4) Φ is parageometric if and only if ρ is not a closed path, ρ traverses some edgeof G exactly once, and ρ traverses some edge of G at least thrice. This occursif and only if K is not a surface and some edge of G is a free edge of K.

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In case (2) we say that g : G → G is Nielsen unique. Also, for the rest of thepaper we use the constant L to represent L(G), so L(ρ) = 2L and ρ : [0, 2L] → G inthe Nielsen unique case.

Proof. Start with any train track representative g′ : G′ → G′ of Φ. From [BFH00]Lemma 4.2.5, g′ has only finitely many periodic Nielsen paths. Pass to a power so thatall Nielsen paths of g′ are fixed. In [BH92], the theory of stable train track represen-tatives is developed, leading to the construction of another train track representativeg : G→ G such that one of the following statements holds:

(1) g has no fixed Nielsen paths.

(2) g has a unique fixed Nielsen path ρ up to orientation reversal, the image of ρ isall of G, L(ρ) = 2L(G), and the illegal turn in ρ is the unique illegal turn of g.

For these statements see in particular Lemma 3.9 of [BH92], and in particular for theproof that the image of ρ is all of G see the top of page 28 of [BH92].

In both of the above statements we need to replace the word “fixed” by the word“periodic”. Section 3 of [BH92] produces g from g′ using a stabilization process, theoutput of which is a sequence of train track representatives g′ = g0, g1, . . . , gN = gso that for each n = 1, . . . , N one of two possibilities holds: gn is obtained fromgn−1 by a Stallings fold, in which case the periodic Nielsen paths of gn−1 and gnare in one-to-one period preserving correspondence; or gn is obtained from gn−1 byeliminating one fixed Nielsen path, and the remaining periodic Nielsen paths of gn−1

are in one-to-one period preserving correspondence with the periodic Nielsen pathsof gn. Since all periodic Nielsen paths of g′ are fixed, the same is true for each gn.Since gN = g has at most one fixed Nielsen path, it has at most one periodic Nielsenpath, which if it exists is fixed.

To complete (1) and (2) we prove the characterization of T+ being geometric,following [BF]. Consider the sequence in Xr defined by [G0] = [G], [Gi] = [Gi−1]Φ. Asa CW-complex, Gi is isomorphic to G. The marking Rr → Gi is obtained inductivelyby postcomposing the marking Rr → Gi−1 with the map g, and so g induces amap gi : Gi → Gi+1 that respects markings (up to homotopy). The metric on Gi isdefined inductively so that gi maps each edge locally isometrically to Gi+1, and soL(Gi+1) = L(Gi)/λ(Φ). The given CW isomorphism ψi : G → Gi has the propertythat Lebesgue measure on Gi is λ(Φ)

−i times the pushforward of Lebesgue measureon G. Also, the following composition equals gi:

G = G0g0−→ G1

g1−→ · · ·

gi−1−−−→ Gi

ψ−1i−−→ G

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Lifting to universal covers one obtains a sequence of Fr-trees and surjective equivari-ant maps between them

T0g0−→ T1

g1−→ T2

g2−→ · · ·

such that gi maps each edge of Ti isometrically onto an arc of Ti+1, and so gi isdistance nonincreasing. Since [Ti] = [Ti−1]Φ and since T+ is the attracting point ofsource-sink dynamics for Φ, it follows that limi→∞ Ti = T+ in X r.

Let T# denote the direct limit of the trees Ti. Set theoretically, this is the set ofequivalence classes of the disjoint union of T0, T1, T2, . . ., where xi ∈ Ti is equivalentto xj ∈ Tj if there exists k ≥ i, j such that xi, xj have the same image in Tk, thatis, gk−1 ◦ · · · ◦ gi(xi) = gk−1 ◦ · · · ◦ gj(xj). Let [xi] ∈ T# denote the equivalence classof xi ∈ Ti. By surjectivity of the maps gi, for each i each equivalence class has theform [xi] for (at least) some xi ∈ Ti. Define a semimetric on T# by the formuladT#([x0], [y0]) = limi→∞ dTi(xi, yi) where by induction we define xi = gi−1(xi−1) andsimilarly for yi; the formula is clearly well-defined independent of the choice of arepresentative in T0 of a given point in T#, and the limit exists because dTi(xi, yi)is a nonincreasing sequence of positive numbers. The actions of Fr on the trees Tiinduce an isometric action of Fr on T#. The map xi → [xi] is an Fr-equivariant

surjective function q#i : Ti → T#. This argument shows that T# is in fact the direct

limit of the sequence T0g0−→ T1

g1−→ · · · , in the category of semimetric spaces with an

isometric Fr-action and distance nonincreasing maps which are Fr-equivariant.In order to understand T# more precisely we need a result from [BFH00]. Fix

i ≥ 0 and an immersed arc or circle γi in Gi, and inductively define γj+1 = (gj ◦ γj)#for j ≥ i. The number of illegal turns in γj is nondecreasing as a function of j andso eventually stabilizes for j ≥ J .

Lemma 6 (Lemma 4.2.6 of [BFH00]). With the notation as above, for each j ≥ Jthe immersion γj is a legal concatenation of legal paths and Nielsen paths. ♦

By a “legal concatenation” we mean that the turn at each concatenation point isa legal turn.

No Nielsen path: strong convergence. Suppose that g has no Nielsen path.Using the above lemma together with the nonexistence of Nielsen paths it followsthat, with the notation as in the lemma, the immersion γj is legal for all sufficientlylarge j. This has several consequences.

First, by applying the above lemma to (the projection to Gi of) the geodesicbetween any two points in any Ti it follows that the semimetric on T# is a metric.Knowing this, it follows that T# is an R-tree metric, because the defining conditionsfor an R-tree metric are closed conditions [CM87]. Next, by applying the abovelemma to loops in Gi it follows that the sequence of translation distance functions

15

for the Fr-trees Ti converges to the translation distance function on T#, from whichit immediately follows that T# = T+. Moreover, applying the lemma again to thegeodesic γi between any two points in Ti, for sufficiently large j the path γj is legal

and so embeds isometrically in T# = T+ under q#i . But this is precisely the definitionof strong convergence of the sequence of Fr-trees Ti to the Fr-tree T+. By the mainresult of [LP97] it follows that the tree T+ is not geometric.

Nielsen unique: the wedge model. Now suppose that g : G → G has a unique(periodic) irreducible Nielsen path ρ = α ∗ β. The wedge model will be an extensionof the train track map g : G → G to a homotopy equivalence k : K → K, where Kis constructed from G and W as in the statement of Proposition 5. The descriptionof k requires imposing on W the structure of a “wedge”, and from this structure wewill also obtain a measured foliation denoted Fs, called the stable foliation of k, sothat the action of k preserves leaves of Fs and multiplies the transverse measure byλ(Φ). We will exhibit geometricity by proving that T+ is dual to Fs.

Choose locally isometric parameterizations α, β : [0, L] → G. The wedge W =△ABC is the triangle in R2 with vertices A = (0,+1), B = (0,−1), C = (L, 0). Theattaching maps are (x, y) 7→ α(x) for (x, y) ∈ AC, and (x, y) 7→ β(x) for (x, y) ∈ BC.The 2-dimensional cell complex K is obtained by attaching W to G in this manner.The 1-skeleton of K is equal to G union the base AB of W , with identificationsA ∼ α(0) and B ∼ β(0). The dihedral valence of a 1-cell E of K is the total numberof times that E is traversed by the attaching maps of the 2-cells of K; the dihedralvalence equals 1 if and only if E is a free edge of K. Note that AB is a free edgeof K. Also, any edge E of G has dihedral valence equal to the number of timesthat ρ traverses G. By collapsing W from the free edge AB we obtain a deformationretraction of K onto G, thereby identifying π1(K) ≈ π1(G) ≈ Fr.

The measured foliation Fs is induced by the vertical measured foliation on△ABCequipped with the transverse measure |dx|. To check compatibility along the 1-skeleton,observe that for each edge E of G, among the segments of AC ∪ BC that map ontoE, the measures obtained on E by pushing forward |dx| via the attaching map allagree with the usual Lebesgue measure on E.

Now we define the extension k : K → K of g : G → G. Define a subdivisionW = P ∪W ′ as follows. P is the subtrapezoid ofW with one base AB =W ∩{x = 0}and with parallel base W ∩ {x = L/λ(Φ)}. W ′ is the subwedge W ′ =W − P , calledthe collapsed subwedge of W for reasons about to become apparent. Define k totake P onto W , stretching the x-coordinate by λ(Φ), and for 0 ≤ x0 ≤ L/λ(Φ)contracting the y-coordinate of vertical segment P ∩ {x = x0} by a factor of c(x0),where c(x) is the unique affine function satisfying c(0) = 1, c(L/λ(Φ)) = 0. In W ′,for L/λ(Φ) ≤ x0 ≤ L the vertical segmentW ′∩{x = x0} is mapped by k to the point

16

g(α(x0)) = g(β(x0)); the latter equation follows from the definition of an indivisibleNielsen path. This completes the definition of the wedge model k : K → K.

We note a few simple facts about k which will be important in what follows:

• k(x) = g(x) for each x ∈ G, so k is continuous.

• k is a homotopy equivalence of K, and the induced outer automorphism onπ1(K) ≈ Fr is Φ.

• For each x ∈ K, the set k−1(x) is described as follows:

– If x ∈ K −G then k−1(x) is a point in K −G.

– If x ∈ G then k−1(x) is either a point in G or a finite, connected graphwhose edges are a (uniformly) finite union of vertical segments of the sub-wedge W ′.

– Every vertical segment of W is eventually collapsed by some power of k,except for the base AB; this follows because in the Nielsen path ρ = α ∗ β,with α, β : [0, L] → G, for each t ∈ (0, L] there exists i ≥ 1 such thatgi(α(t)) = gi(β(t)).

To understand the description of k−1(x), the map k is defined by collapsing to a pointeach of the vertical segments of the subwedgeW ′. Each nontrivial point pre-image ofk is therefore described by picking some vertical segment ℓ0 of W ′, then inductivelydefining ℓi be the union of ℓi−1 with all vertical segments of W ′ that touch ℓi−1, andthen taking the union of the ℓi to obtain a graph. The vertices of this graph are thepoints of intersection of the graph with G, all of the vertices are identified to a singlepoint under g = k

∣∣ G, but g is uniformly finite-to-one.The inclusion G → K induces a marking of the 2-complex K by the homotopy

equivalence Rr → G → K. LetK0 = K and letKi be the marked 2-complex similarlyobtained from Gi by attaching a wedge along the Nielsen path. The homotopyequivalence k : K → K induces a marking preserving homotopy equivalence ki : Ki →Ki+1 that agrees with gi : Gi → Gi+1. Note that the homeomorphism ψi : G → Giextends to a homeomorphism also denoted ψi : K → Ki such that the composition

K = K0k0−→ K1

k1−→ · · ·ki−1−−−→ Ki

ψ−1i−−→ K equals ki. Letting Fs

0 = Fs, there is ameasure foliation Fs

i defined inductively on Ki as the pushforward of Fsi−1 via the

map ki−1. Note that under the homemorphism ψi : K → Ki, Fsi is the pushforward

of Fs with transverse measure multiplied by λ(Φ)−i. Lifting to universal covers, weobtain a 2-complex Ki containing the tree Ti, and an action of Fr on Ki extendingthe action on Ti. The map gi : Ti → Ti+1 extends to a map ki : Ki → Ki+1 that is alift of ki : Ki → Ki+1. There is an Fr-equivariant measured foliation Fs

i on Ki that

is the lift of Fsi as well as the pushforward of Fs

i−1 via ki−1.

17

Our goal is to prove that the dual Fr-tree of Fs — the metric space associatedto the semimetric on K obtained by integrating Fs along paths — can be identifiedwith T+. We work towards this goal somewhat indirectly, working mostly with thesemimetric space T# instead of the tree T+, and with the leaf space L of Fs insteadof the dual R-tree of Fs.

The semimetric on K, when evaluated on two points of the same leaf of Fs, equalszero, and so there is an induced semimetric on L and there are distance preservingsurjective maps K → L and from L to the dual R-tree of Fs. Define a surjectivemap α : T# → L so that, for x ∈ T0, α[x] is the leaf of Fs passing through x. Tosee that α is well defined, note that for x, y ∈ T0 we have [x] = [y] if and only if x, yhave the same image in some Ti, which occurs if and only if there exists a sequenceof vertical segments of the wedge W connecting x to y each of which are eventuallycollapsed by k, but this implies that x, y are in the same leaf of Fs.

We claim that:

(A) The map α : T# → L is distance preserving.

To prove this, consider x0, x′0 ∈ T0 contained in respective leaves ℓ, ℓ′ of Fs, and let

ξ = [x0], ξ′ = [x′0], so α(ξ) = ℓ and α(ξ′) = ℓ′. Define inductively xi = gi−1(xi−1) and

similarly for x′i. Let γi be the geodesic in Ti between xi and x′i, so the sequence LTi(γ)

is nonincreasing and has limit dT#(ξ, ξ′). Choosing ǫ > 0, for sufficiently large I wehave LTI (γI) ≤ dT#(ξ, ξ′) + ǫ. Let ρI = γI , and inductively define a path ρi in Ki

for i = I − 1, . . . , 0 as follows: at any point where ρi does not pull back continuouslyto Ki−1, one can interpolate a leaf segment of Fs

i−1 to get a continuous path ρi−1 in

Ki−1 connecting xi to x′i, such that

∫ρi−1

Fsi−1 =

∫ρiFsi . We therefore have

∫

ρ0

Fs0 =

∫

ρI

FsI = LTI (γI) < dT#(ξ, ξ′) + ǫ

Letting ǫ → 0, it follows that dL(α(ξ), α(ξ′)) ≤ dT#(ξ, ξ′). For the opposite inequality,

choosing ǫ > 0 let ρ0 be a continuous path in K from x0 to x′0 so that∫ρ0Fs ≤

dT (ℓ, ℓ′) + ǫ/2. Without increasing the integral along ρ0 we may rewrite it as a

concatenation of immersed paths in T0 and vertical segments of wedges. Inductivelydefine the path ρi in Ki as ki−1◦ρi−1, which has the effect of collapsing certain verticalwedge segments of ρi−1, and so L(ρi) = L(ρi−1). By induction, L(ρi) = L(ρ0). Forsufficiently large i, say i ≥ I, all vertical wedge segments of ρ0 have been collapsed inρi except for those which are lifts of the base AB of the wedge W ; let γi be the pathin Ti obtained from ρi be replacing each such vertical segment with the associatedNielsen path. For i ≥ I the number of these Nielsen paths is constant, and theirlength goes to zero as i→ ∞, and so for sufficiently large i we have

dT#(ξ, η) ≤ L(γi) ≤ L(ρi) + ǫ/2 < dT (ℓ, ℓ′) + ǫ

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This proves the claim.We claim next that:

(B) For each ξ, ξ′ ∈ T#, α(ξ) = α(ξ′) if and only if dT#(ξ, ξ′) = 0.

If dT#(ξ, ξ′) 6= 0 then the previous claim shows that dL(α(ξ), α(ξ′)) 6= 0 and so α(ξ) 6=

α(ξ′). To prove the converse, suppose that dT#(ξ, ξ′) = 0. Choosing x0, x′0 ∈ T0 so

that ξ = [x0], ξ′ = [x′0], we must prove that the points x0, x

′0 are contained in the

same leaf of Fs. Define inductively xi = gi−1(xi−1) and similarly for x′i. Let γi bethe geodesic between xi and x

′i in Ti. Applying Lemma 4.2.6 of [BFH00] (Lemma 6

above), for sufficiently large i, say i ≥ I, the path γi is a legal concatenation of legalpaths and Nielsen paths. As i ≥ I increases, the lengths of the legal paths in γi staythe same while the number of Nielsen paths is constant and their lengths go to zero.It follows that dT#(ξ, ξ′) = 0 only if γi has no legal paths for i ≥ I, that is, γI is aconcatenation of Nielsen paths. We may therefore connect xI to x′I by a path ρI in

KI entirely contained in a leaf of FsI . Now proceeding inductively as in the earlier

claim, for i = I, I − 1, . . . , 0 we obtain a path ρi in Ki entirely contained in a leaf ofFsi connecting xi to x

′i, and taking i = 0 it follows that α(ξ) = α(ξ′).

From the above two claims we can draw the conclusion that the semi-metricon L is a metric: if ℓ, ℓ′ ∈ L and dL(ℓ, ℓ

′) = 0 then, choosing ξ ∈ α−1(ℓ) andξ′ ∈ α−1(ℓ′), Claim (A) implies that dT#(ξ, ξ′) = 0, and Claim (B) implies thatℓ = α(ξ) = α(ξ′) = ℓ′. It immediately follows, follows from the definition of L, thatL is equal to the associated metric space of the semimetric on K, in other words, Lis equal to the dual tree of Fs. It also immediately follows that the map α identifiesL with the metric space associated to the semimetric on T#. Moreover, we knowexactly which distinct pairs of points in T# have distance zero, namely, those pairsξ, ξ′ represented by x0, x

′0 ∈ T0 contained in the same leaf ℓ of Fs

0 so that x0 and x′0are separated from each other in ℓ by the lifts of AB in ℓ.

To complete the proof that T+ is geometric, we must check that the translationdistance function of the Fr-tree L is the limit of the translation distance functions ofthe Fr-trees Ti, for that will identify L with T+. Consider c ∈ C represented by animmersed closed curve γ0 in G0. Inductively define γi = (gi ◦ γi−1)#, so LGi

(γi) isthe translation length of c in Ti. By Lemma 6, for sufficiently large i, say i ≥ I, γi isa legal concatenation of legal paths and Nielsen paths, where the number of Nielsenpaths is a constant independent of i and their lengths go to zero, so limLGi

(γi)is equal to the total length of the portion of γi which is not in one of the Nielsenpaths, the latter number being independent of i. Let Ai ⊂ Ti be an axis of (anyrepresentative of) c acting on Ti. The map Ai → T# → L = T+ has the effect offolding each Nielsen path in Ai into a segment in T#, and mapping the rest of Aionto the axis A+ of c in T+. It follows that a fundamental domain for A+ has length

19

equal to the length of a fundamental domain for Ai minus the Nielsen paths, whichequals limLGi

(γi).

This completes the proofs of items (1) and (2) of Proposition 5. Item (3) wasproved in [BH92]. It immediately follows that Φ is parageometric if and only if ρ isnot closed. Moreover, in this case, if ρ traverses every edge at least twice then usingL(ρ) = 2L it follows that ρ traverses every edge exactly twice, implying that Φ isgeometric, a contradiction. It follows that ρ traverses some edge E at most once. Onthe other hand, by item (2) ρ traverses each edge at least once, and so ρ traversesE exactly once, implying that E is a free edge of K. Using again that L(ρ) = 2L itfollows that ρ traverses some other edge at least thrice. This proves item (4). ♦

3 The stable foliation of the wedge model

In this section we study the stable foliation Fs of a wedge model k : K → K for Φ.The main result, Proposition 15, says that the set of bi-infinite lines in Fs, calledthe hull of Fs, can be identified with the leaves of the expanding lamination of Φ−1.The results of this section are closely related to results found in [BF]. Indeed, webelieve that Proposition 15 can be proved by the Rips machine methods of [BF], butwe have developed a different proof.

3.1 The stable foliation

For each x ∈ K the leaf of Fs through x is called the stable leaf of x, denoted Fsx.

We can build Fsx up inductively: let ℓ0 = x, define ℓi inductively as the union of

ℓi−1 with all vertical segments of W that intersect ℓi−1, and then Fsx = ∪iℓi. Note

that Fsx is a locally finite, connected 1-complex with vertex set Fs

x ∩ G and whoseedges are vertical segments of the wedge W . Any p, q ∈ Fs

x ∩G are connected by anedge path in Fs

x consisting of the concatenation of a consecutive sequence of verticalsegments of W going from p to q. As we will see in Fact 10 below, each leaf Fs

x

is a tree, so the minimal edge path [p, q] between vertices p, q ∈ Fsx is unique, and

the number of edges on this path is denoted LK [p, q], called the leafwise distancebetween p and q. Notice that we do not measure LK using lengths of segments inthe Euclidean triangle model of the wedge W . One should beware that the leafwisedistance between p and q is not a continuous function, when regarded as a functionof the ordered pairs (p, q) ∈ G×G such that p, q are contained a common leaf of Fs

x,because as p, q vary in G the edge path [p, q] could vary in such a way that it passesover the apex of W where the leafwise distance jumps discontinuously.

For each x ∈ G the valence of the leaf Fsx at x equals the number of times that

ρ passes over x except at an endpoint or the midpoint of the domain of ρ. If x is in

20

the interior of an edge E of K, this number is just the dihedral valence of E in K.Since there are only finitely many vertices, it follows that there is a uniform upperbound for the valences of all vertices in all leaves of Fs

x.One point of confusion is the fact that, near the apex of the wedgeW , the vertical

segments of W get shorter and shorter, and it may seem possible that such segmentscould accumulate in a leaf of Fs

x. This is not possible, however, because of localfiniteness of the complex K: if v is the vertex of K to which the apex of W isidentified, and if U is a small neighborhood of v in K, then for every vertical segmentα of W contained in U , and for every vertical segment α′ of W such that α and α′

share a common endpoint, the segment α′ is not contained in U ; see also the end ofthe proof of Lemma 9. In fact one sees that the path topology on each leaf Fs

x, whichhas as basis the path components of Fs

x ∩U over all open subsets U ⊂ K, is the sameas the CW-topology on the simplicial complex Fs

x.From the definition of Fs and the map k it is clear that k preserves the foliation Fs,

mapping each leaf Fsx onto the leaf Fs

k(x), inducing a bijection of leaves. The next

fact (almost) justifies the terminology “stable foliation” for Fs. Define the strongstable set of x ∈ K to be

Fssx = {y ∈ K

∣∣ ∃i ≥ 0 s.t. ki(x) = ki(y)}

Fact 7 (The stable foliation). Two points x, y ∈ K are in the same leaf of Fs ifand only if there exists i ≥ 0 such that ki(x) = ki(y) or ki(x), ki(y) ∈ AB. It followsthat for each x ∈ K we have:

(1) If AB 6⊂ Fsx then Fss

x = Fsx.

(2) If AB ⊂ Fsx then the following hold.

(a) If x 6∈ int(AB) then Fssx is the component of Fs

x− int(AB) that contains x.

(b) If x ∈ int(AB) then Fssx = {x}.

Proof. This follows immediately from the observation that for any vertical segment γof the wedgeW , there is a power of k collapsing γ to a point if and only if γ 6= AB. ♦

Fact 8 (Stable leaves are infinite). Each leaf of Fs is an infinite 1-complex.

Proof. By Fact 7, for each x ∈ G and each n ≥ 0, the vertex set of the leaf of Fs

through x contains the set g−n(gn(x)), whose cardinality goes to infinity as n → ∞since the transition matrix of g is Perron–Frobenius. ♦

21

3.2 The structure of stable leaves

Consider the universal covering spaces G ⊂ K. Let Fs be the foliation of K obtainedby lifting Fs.

Given a leaf ℓ of Fs, we study the structure of ℓ by considering a locally embeddedfinite edge path p in ℓ. Since p is an edge path, it starts and ends on G. Let thesequence of points of intersection of p with G be denoted x0, x1, . . . , xJ . For example,one question we want to answer is whether ℓ is a tree, which is true if and only ifx0 6= xJ for all nontrivial edge paths p in ℓ; see Fact 10. Another question is whetherℓ is quasi-isometrically embedded in K, which is true if and only if the map i 7→ xiis a quasi-isometric embedding Z → K; see Fact 12.

We establish further notation regarding p. For each j = 1, . . . , J , the segmentof the path p from xj−1 to xj is a vertical segment in a lift of the wedge, and so

there exists a lift ρj : [0, 2L] → G of ρ or ρ, and there exists tj ∈ [0, L), such that

xj−1 = ρj(L − tj) and xj = ρj(L + tj). Let Vj = ρj(L) ∈ G be the point at which

ρj makes its unique illegal turn. The geodesic xj−1 xj in G is a path of length 2tjobtained from ρj by truncating the initial and final segments of length L − tj, andso xj−1 xj = αj ∗ βj concatenated at the illegal turn at Vj with αj , βj legal paths oflength tj.

As a final remark, note in the discussion above that for each j = 1, . . . , J − 1,since the subpath of p from xj−1 to xj+1 in ℓ is locally embedded, the vertical wedgesegments from xj−1 to xj and from xj to xj+1 are not inverses of each other, and so

the paths αj ∗ βj and αj+1 ∗ βj+1 are not inverses of each other in G. Since G has

a unique illegal turn, each point of G at which an illegal turn occurs has a uniqueillegal turn, from which it follows that Vj 6= Vj+1.

Lemma 9. Using the notation above, for each locally embedded edge path p in a leafof Fs and for each j = 0, . . . , J there exists i with 1 ≤ i ≤ J such that the geodesicx0 xJ makes an illegal turn at Vi, and the path xj Vi is legal. Moreover, L(xj Vi) ≤ L.

Proof. We will prove the final statement about length at the very last. The proof ofthe rest of the lemma is by induction on J , with J = 1 obvious. Supposing that thelemma is true for a certain J , we wish to prove it for J + 1.

Choose xj with 0 ≤ j ≤ J + 1. We may assume that j ≤ J , because if j = J + 1then we can just reverse the direction of p, reducing to the case j = 0. We thereforecan apply the induction hypothesis to the subpath of p from x0 to xJ , obtaining kwith 1 ≤ k ≤ J such that x0 xJ has an illegal turn at Vk.

Decompose x0 xJ = µ ∗ ν at the turn Vk; no claims are made on the legality of µor ν. The path x0 xJ+1 is what you get by tightening µ ∗ ν ∗ αJ+1 ∗ βJ+1, which isdone by cancelling a terminal segment of µ ∗ ν with an initial segment of αJ ∗ βJ .

22

If the cancellation does not remove all of ν then Vk is still a turn in x0 xJ+1 andwe are done, by taking Vi = Vk. So assume that all of ν cancels. Since αJ+1 is legalit does not cancel with any of µ.

If not all of αJ+1 cancels with ν then x0 xJ+1 takes the illegal turn VJ+1. To showthat the path xi VJ+1 is legal, this path is a concatenation of the legal path xj Vk withthe path Vk VJ+1 which is legal because it is a subpath of αJ+1, and by constructionthe turn at which these two paths are concatenated is not the illegal turn at the pointVk, and so this turn is legal. We are therefore done, taking Vi = VJ+1.

The remaining case is when ν = αJ+1. In this case Vk = VJ+1 and so the illegalturns of x0 xJ = µ ∗ ν and αJ+1 ∗ βJ+1 at the common concatenation point are thesame, because G has at most one illegal turn at each vertex. Also, the Nielsen pathsρJ+1, ρk are equal up to orientation, because each is a lift of ρ or ρ and they havethe same illegal turn. This implies that (J + 1) − k ≥ 2, from the final remark justbefore Lemma 9; this is where we use the hypothesis that p is locally embedded. Ifthe orientations of the Nielsen paths ρk and ρJ+1 agree, then the points xk and xJ+1

lie on the same legal half of this Nielsen path, and so the geodesic xkxJ+1 is legal.But this geodesic falls under the induction hypothesis because 1 ≤ J + 1 − k ≤ J ,and so xkxJ+1 must have an illegal turn, a contradiction. If ρk, ρJ+1 are oriented inthe opposite direction, then the path xkxJ is legal, which also violates the inductionhypothesis, noting that 1 ≤ J − k ≤ J . This completes the induction.

To prove that xj Vi has length ≤ L, choose n and a lift gn : G→ G of gn : G→ Gso that gn(xj) = gn(xi). The legal paths xj Vi and xi Vi being identified under gn,their lengths are equal. But

L(xi Vi) ≤1

2L(ρi) =

1

2L(ρ) = L

♦

3.3 Consequences of Lemma 9

The first consequence is:

Fact 10. Each leaf of Fs is a tree.

Proof. Lemma 9 shows that for each locally embedded edge path in a leaf of Fs, thegeodesic in G with the same endpoints has an illegal turn, and so the endpoints aredistinct. This implies that the leaf has no loops and so is a tree. ♦

Fact 11. Each leaf of Fs is a tree.

Proof. Let ℓ be a leaf of Fs. By Fact 10, ℓ is π1-injective in K. If ℓ were not atree then then ℓ would contain an embedded loop. By Fact 7 the image of this loop

23

under a sufficiently high power of the homotopy equivalence k is either a point or thesegment AB, violating π1-injectivity. ♦

Consider now any Fr-equivariant proper geodesic metric on K, and any leaf ℓ ofFs, with the simplicial structure on ℓ inducing a simplicial metric where each edgehas length 1. Next we essentially prove that the injection ℓ→ K is a quasi-isometricembedding, with quasi-isometry constants independent of ℓ. What we actually need,and prove, is an interpretation of this statement that takes place entirely in G:

Fact 12. For each leaf ℓ of Fs, and each x, y ∈ ℓ ∩ G, letting x = x0, . . . , xJ = y bethe points of ℓ∩ G from x to y in order, the map i 7→ xi is a quasigeodesic embeddingof {0, 1, . . . , J} into G, with quasigeodesic constants independent of x, y.

Proof. Let L(·) denote length in G. Since J is arbitrary, it suffices to prove thatL(x0 xJ) is bounded above and below by an affine function of J . Since L(xi xi+1) ≤ 2Lwe have L(x0 xJ) ≤ 2L · J .

Lemma 9 gives a map f from the set {x0, . . . , xJ} to the set of illegal turns of thegeodesic x0 xJ , such that d(xj , f(xj)) ≤ L. There is an integer κ ≥ 1 such that theL neighborhood of each illegal turn Vi intersects at most κ different edges. Lemma 9also implies that each edge contains at most one of the points {x0, . . . , xJ}, and sothe L neighborhood of Vi contains at most κ of the points {x0, . . . , xJ}. This showsthat the map f is at most κ-to-one, and so its image has cardinality ≥ J+1

κ . Inother words, the geodesic x0 xJ has at least J+1

κ illegal turns. Letting η > 0 be theminimum length of an edge of G, it follows that

L(x0 xJ) ≥ (J + 1

κ− 1)η =

η

κJ + η(

1

κ− 1)

♦

3.4 The hull of the stable foliation

The hull of a simplicial tree is the union of bi-infinite lines in the tree, and the hull ofFs is the union of the hulls of its leaves, a closed foliated subset of K denoted H(Fs).The hull of Fs is similarly defined, and H(Fs) is equal to the total lift of H(Fs).By Fact 12, each bi-infinite line in a leaf of H(Fs) intersects G in a quasigeodesicembedding Z → G and so has finite Hausdorff distance from a bi-infinite line in G.Our goal in this section is to identify this collection of lines with a natural extensionof the lamination Λu(Φ−1).

Fact 13. H(Fs) is a nonempty, closed lamination, using the compact-open topologyon maps R → K. For each leaf ℓ of HFs, k(ℓ) is also a leaf of HFs, and this inducesa bijection of the set of leaves of HFs. The leaf ℓ′ that maps to ℓ is the unique leafcontained in the set k−1(ℓ).

24

Proof. As we saw in Fact 8, each leaf has infinitely many vertices. Also, each vertexhas finite valence — in fact, the valence of each vertex of each leaf is uniformlybounded, by the maximum cardinality of ρ−1(x) for x ∈ G. It follows that eachleaf contains an infinite ray, and so there exists a sequence of longer and longer leafsegments αi centered on a sequence of points xi. Any limit point x of the sequence xilies on a bi-infinite line in some leaf. This shows simultaneously that H(Fs) is closedand that it is nonempty.

The map k is finite-to-one on G, and it extends to K by collapsing to a pointeach vertical segment contained in the collapsed subwedge W ′ of W . It follows thateach point pre-image is a connected subset of a leaf of K consisting of a union ofboundedly many edges in that leaf, and since each leaf is a tree (by Fact 11), eachpoint pre-image is a finite subtree. The intersection of ℓ with a point pre-image istherefore a finite subarc. In other words, the effect of k on ℓ is to collapse to a pointeach of a pairwise disjoint collection of uniformly finite subarcs of ℓ, which impliesthat k(ℓ) is also a bi-infinite line in a leaf of Fs. Thus k induces a well-defined self-map of the set of leaves of HFs. This map is injective, because for each leaf ℓ ofHFs, the set k−1(ℓ) maps to ℓ by collapsing a pairwise disjoint collection of uniformlybounded finite subtrees, and so k−1(ℓ) contains a unique leaf ℓ′ of H(Fs). Since k(ℓ′)is contained in ℓ and equals a leaf of HFs, it follows that k(ℓ′) = ℓ, showing that themap is surjective. ♦

Now we set up notation for pushing leaves of H(Fs) from K into the graph G.Consider a leaf ℓ of H(Fs). Picking a base point x0 ∈ ℓ ∩ G and an orientation of ℓdetermines an ordering of the set ℓ ∩ G, giving a bi-infinite sequence that we denotex(ℓ) = (. . . , x−1, x0, x1, . . .). This sequence has the following properties:

(1) For each i, the geodesic xi−1 xi in G is a subpath of a lift ρ = α ∗ ˜β of the

Nielsen path ρ, consisting of equal length terminal subpaths of α and ˜β.

(2) The sequence x(ℓ) is one-to-one.

Conversely, any sequence x = (. . . , x−1, x0, x1, . . .) satisfying (1) and (2) is equal tox(ℓ) for some leaf ℓ of H(Fs) and some base point and orientation of ℓ.

Given a leaf ℓ of H(Fs) and x(ℓ) = (. . . , x−1, x0, x1, . . .) as above, by Fact 12the sequence x(ℓ) is a quasi-isometric embedding Z → G with uniform constantsindependent of ℓ. It follows that x(ℓ) fellow travels a unique bi-infinite line in Gdenoted ℓ. We thus obtain sublaminations of the geodesic laminations of G and of G:

ΛH = {ℓ∣∣ ℓ is a leaf of H(Fs)} ⊂ ΛG

ΛH = the projection of ΛH to ΛG

25

For a sequence x satisfying (1) and (2) above, the image of x under g also has thestructure of a sequence satisfying (1) and (2), and g induces a quasi-isometry from xto g(x). It follows that g induces a quasi-isometry ℓ→ ℓ′ well-defined up to boundeddistance. We may therefore set Φ(ℓ) = ℓ′, which induces a map ΛH → ΛH downstairs,that is, ΛH is an invariant sublamination of the action of Φ on the geodesic laminationof G.

A leaf ℓ of the geodesic lamination of Fr is birecurrent if its realization in any(equivalently, some) marked graph has the property that each finite subpath occursinfinitely often in both ends of ℓ. Equivalently, ℓ is contained in the set of limits pointsof each of its ends, where the convergence takes place in the geodesic lamination.Compactness of K implies that H(Fs) is compact and so, as a consequence of theHausdorff maximum principle, H(Fs) has a nonempty minimal sublamination. Eachleaf of a minimal sublamination is birecurrent. We therefore have shown:

Fact 14. ΛH contains a birecurrent leaf. ♦

Here is the main result of this section. Following the proof we will strengthen itby sketching how to identify ΛH completely. Recall that Λu(Φ) denotes the expand-ing lamination of Φ and λu(Φ−1) is the expanding lamination of Φ−1. In the nextproposition we consider the realizations of each of these laminations in the train trackmap g : G→ G representing Φ.

Proposition 15. ΛH ⊃ Λu(Φ−1).

Proof. First we show that there is a uniform bound to the length of any legal subpathof any leaf of ΛH. To see why, fix a leaf ℓ of ΛH, and let x(ℓ) = (. . . , x−1, x0, x1, . . .).For each i 6= j, each point of the geodesic xi xj has distance ≤ L from some xk withi ≤ k ≤ j, and by applying Lemma 9 it follows that xk has distance ≤ L from someillegal turn of xi xj, so the illegal turns on xi xj are spaced no more than 4L apart.Any finite subsegment of ℓ of length > 4L is contained in some xi xj and so containsan illegal turn.

Recall from [BFH00] that a leaf ℓ of the geodesic lamination is weakly attractedto Λu(Φ) if any finite subpath of a leaf of Λu(Φ) is a subpath of gn(ℓ) for somelarge n. Since the subpath can be a legal segment of arbitrarily large length, itfollows that gn(ℓ) contains legal paths of arbitrarily large length. But we have shownthat if ℓ ∈ ΛH then gn(ℓ) ∈ ΛH and so there is an upper bound to the length oflegal subpaths. It follows that no leaf of ΛH is weakly attracted to Λu(Φ). ApplyingTheorem 6.0.1 of [BFH00], it follows that each birecurrent leaf of ΛH is a leaf ofΛu(Φ−1), so there exists at least one leaf of Λu(Φ−1) that is contained in ΛH. Byminimality of Λu(Φ−1) it follows that Λu(Φ−1) ⊂ ΛH. ♦

26

For completeness sake we give a description of ΛH, but with details of proof onlysketched lightly since we do not need this description for our present purposes. We’llassume that g : G → G is a train track representative of Φ−1 which is either Nielsenunique or has no Nielsen path at all. The extended expanding lamination of Φ−1 isdefined to be the union of Λu(Φ−1) with finitely many other leaves, as follows. At aperiodic vertex p of g, any periodic direction d determines a ray rd under iteration ofg, and any two periodic directions d, d′ determine a leaf ℓ = rd ∪ rd′ in the extendedexpanding lamination. Also, if ρ is the Nielsen path connecting two points p and q,then for any periodic directions d, d′ at p, q respectively, distinct from the directionsof ρ at its endpoints, ℓ = rd ∪ ρ ∪ rd′ is a leaf in the extended expanding lamination.A study of the stabilization algorithm of [BH92] shows that the extended expandinglamination is well-defined, independent of the choice of g.

We claim that ΛH is the extended expanding lamination of Φ−1. This is a con-sequence of the proof of Theorem 6.0.1 of [BFH00] applied to a train track represen-tative g : G → G for Φ. This proof is laid out with Steps 1, 2, and 3. In the endof Step 2, one considers a geodesic ℓ, for example any leaf of ΛH, whose realizationin G is not weakly attracted to Λu(Φ), meaning that the sequence (giℓ)# does notdevelop longer and longer segments that are leaf segments of Λu(Φ). What one showsin this situation is that there exists an immersed loop γ in G of uniformly boundedlength so that the sequence γi = (giγ)# develops longer and longer segments that arecontained in and exhaust ℓ, as realized in G. Applying Lemma 6, for some I the loopγI is a legal concatenation of legal paths and Nielsen paths, and it follows that fori ≥ I the loop γi is a legal concatenation of a uniformly bounded number of segmentseach of which is either a leaf segment of Λu(Φ−1) or of one of the finitely many leavesadded to make the extended expanding lamination of Φ−1. One then sees that ℓ isexhausted by such segments, implying that ℓ itself is a leaf of the extended expandinglamination.

4 Proof of the main theorem

In this section we use LK(α) to denote combinatorial length of an edge path α in aleaf of Fs or Fs, that is, the number of vertical wedge segments in α. We also useLG to denote length of a path in the graph G or G.

Let G1 be the subgraph of G consisting of the edges of dihedral valence > 1 in K.Let I1 = {x ∈ G

∣∣ kn(x) ∈ G1 for all n ≥ 0}, so I1 is a Cantor set in G1.

Fact 16. For any leaf ℓ of ΛH, a bi-infinite line in a leaf of Fs, we have

ℓ ∩G ⊂ I1

27

Proof. Clearly ℓ ∩ G ⊂ G1 for each bi-infinite line in a leaf of Fs, because for eachpoint x ∈ G that is not a vertex, the valence of x in its Fs leaf equals the dihedralvalence of the edge containing x. For each n ≥ 0, since kn(ℓ) is a bi-infinite line in aleaf of Fs, it follows that kn(ℓ) ∩G ⊂ G1, and therefore ℓ ∩G ⊂ I1. ♦

Our next fact gives the strict inequality that we shall need in proving thatλ(Φ−1) < λ(Φ).

Fact 17. There is a number λ′ < λ(Φ) such that for each x ∈ I1 we have

lim supn→∞

1

nlog

∣∣g−n(x) ∩ I1∣∣ ≤ log(λ′)

Proof. Recall that Γ is the transition graph of G. Let Γ1 be the subgraph obtainedfrom Γ by throwing away each vertex associated to an edge of G of dihedral valence= 1, and any directed edge of Γ incident to such a vertex. Let M1 be the transitionmatrix of Γ1. If x is contained in the interior of the ith edge of G, then |g−n(x) ∩ I1| isthe sum of the entries in the ith column of Mn

1 . In all cases |g−n(x) ∩ I1| is boundedabove by |Mn

1 |, the sum of all coefficients in Mn1 . It therefore suffices to show that

lim supn→∞

1

nlog |Mn

1 | ≤ log(λ′) < log(λ).

Since we may regard M1 as defined on the same set E × E as M with M1(e, e′) ≤

M(e, e′) and with strict inequality for at least one pair (e, e′) ∈ E × E , and since λ isthe Perron-Frobenius eigenvalue of M , this inequality follows from Perron-Frobeniustheory. See for example Theorem 4.4.7 and Theorem 4.4.4 of [LM95]. ♦

Now we relate Fact 17 to the asymptotic compression factor of k acting on leavesof HFs.

Fact 18. For each ǫ > 0 there exists an integer N ≥ 1 such that for all n ≥ N , if αis an arc in a leaf ℓ of HFs, and if LK(α) is sufficiently long (depending on n), then

LK(α)

LK(knα)≤ (λ′ + ǫ)n

Proof. Applying Fact 13, for each leaf ℓ of HFs let k(ℓ) denote the unique leaf ofHFs such that k(k(ℓ)) = ℓ, so k(ℓ) ⊂ k−1(ℓ). We extend the set map k to subsets ofleaves of HFs, by setting k(α) = k(ℓ) ∩ k−1(α) for each α ⊂ ℓ.

Let ℓ be a leaf of HFs and α ⊂ ℓ an arc. Let knα ∩ G = {x0, x1, . . . , xI} inorder. Note that if 0 < i < I then kn(xi) ⊂ α, so we can augment α, replacingit by kn(x0) ∪ α ∪ kn(xI), and increasing the length of α by an amount depending

28

only on n, without changing knα. If LK(α) were sufficiently long to start with thenthis increase would change LK(α) by an arbitrarily small factor. We may thereforeassume that α = kn(knα).

It follows that α is equal to the disjoint union of the arcs kn(x0), . . . , kn(xI)

together with I additional open vertical wedge segments, each of which maps home-omorphically by kn to the I open wedge segments that constitute the arc knα. Wetherefore have

LK(α) =

I∑

i=0

LK(kn(xi)) + I

and dividing by LK(knα) = I we get

LK(α)

LK(knα)=

1

I

I∑

i=0

LK(kn(xi)) + 1

We also have LK(kn(xi)) ≤ |g−n(xi) ∩ I1|. By combining this with Fact 17 it follows

that if n is sufficiently large then LK(kn(xi)) ≤ (λ′ + ǫ

2)n and so

LK(α)

LK(knα)≤

(λ′ +

ǫ

2

)n+λ′ + ǫ

2

I+ 1

By taking n sufficiently large, this last quantity is

≤ (λ′ + ǫ)n

♦

Now we prove the main theorem. Choose a train track representative γ : Γ → Γof Φ−1 and let γ : Γ → Γ be a lift to the universal cover. Since g : G → G is atrain track representative of Φ, we can choose the lifts γ : Γ → Γ and g : G → Gto represent inverse automorphisms of Fr. It follows that there is an Fr-equivariantquasi-isometry h : G → Γ such that h is a “quasiconjugacy” between k and “γ−1”,meaning that d(γ ◦h◦ g(x), h(x)) is uniformly bounded over all x ∈ G. Now we applythis to a particular leaf of Λu(Φ−1) as follows. Pick a leaf ℓ′ of Λu(Φ−1) realized in Γ.For convenience we assume ℓ′ is periodic under γ. By passing to a power of Φ wemay assume that ℓ′ is fixed by γ. We may choose a lift ℓ′ ⊂ Γ, and we may choosethe lifts g and γ, so that ℓ′ is fixed by γ. Let ℓ denote the corresponding leaf of HFs,and so ℓ is fixed by k. Pushing ℓ into G, and mapping over by h to Γ, the result isHausdorff equivalent to ℓ′, and composing by the closest point projection to ℓ′, we

29

obtain a map still denoted h : ℓ→ ℓ′ which is a quasi-isometry and a quasiconjugacy,that is, there are constants κ ≥ 1, η ≥ 0 such that

1

κdℓ(x, y)− η ≤ dℓ′(hx, hy) ≤ κdℓ(x, y) + η, for all x, y ∈ ℓ

and such thatdℓ′

(γ ◦ h ◦ k(x), h(x)

)≤ η for all x ∈ ℓ.

Here we use dℓ for distnce in ℓ and dℓ′ for distance in ℓ′. By induction we obtain foreach n a constant ηn ≥ 0 such that

dℓ′(γn ◦ h ◦ kn(x), h(x)

)≤ ηn, for all x ∈ ℓ′

We know that γ expands length on ℓ′ by the exact factor of λ(Φ−1). On the otherhand, Fact 18 tells us that k contracts length on ℓ by an asymptotic factor of atmost λ′. Combining these, we now show that λ(Φ−1) ≤ λ′.

By Fact 18, for each sufficiently long leaf segment [x, y] of ℓ we have

dℓ(x, y) ≤ (λ′ + ǫ)n dℓ(kn(x), kn(y))

and we also have

dℓ(kn(x), kn(y)) ≤ κdℓ′(h ◦ kn(x), h ◦ kn(y)) + κη

dℓ′(h ◦ kn(x), h ◦ kn(y)) = λ(Φ−1)−ndℓ′(γn ◦ h ◦ kn(x), γn ◦ h ◦ kn(y))

dℓ′(γn ◦ h ◦ kn(x), γn ◦ h ◦ kn(y)) ≤ dℓ′(hx, hy) + 2ηn

dℓ′(hx, hy) ≤ κdℓ(x, y) + η

Combining these we get

dℓ(x, y) ≤ (λ′ + ǫ)nλ(Φ−1)−nκ2dℓ(x, y) + (λ′ + ǫ)n(η + 2ηn + κη)︸ ︷︷ ︸Bn

1 ≤ (λ′ + ǫ)nλ(Φ−1)−nκ2 +Bn

dℓ(x, y)

By Fact 18, we can let dℓ(x, y) → ∞ and we get

λ(Φ−1)n ≤ (λ′ + ǫ)nκ2

λ(Φ−1) ≤ (λ′ + ǫ)κ2/n

30

and then we can let n→ ∞ to get

λ(Φ−1) ≤ λ′ + ǫ

Finally, letting ǫ→ 0 we get

λ(Φ−1) ≤ λ′ < λ(Φ)

This completes the proof of the main theorem.

References

[BF] M. Bestvina and M. Feighn, Outer limits, preprint.

[BF95] M. Bestvina and M. Feighn, Stable actions of groups on real trees, Invent.Math. 121 (1995), no. 2, 287–321.

[BFH97] M. Bestvina, M. Feighn, and M. Handel, Laminations, trees, and irre-ducible automorphisms of free groups, Geom. Funct. Anal. 7 (1997), 215–244.

[BFH00] M. Bestvina, M. Feighn, and M. Handel, The Tits alternative for Out(Fn).I. Dynamics of exponentially-growing automorphisms., Ann. of Math. 151(2000), no. 2, 517–623.

[BFH04] M. Bestvina, M. Feighn, and M. Handel, The Tits alternative for Out(Fn).II. The Kolchin theorem., Ann. of Math. (2004), to appear.

[BH92] M. Bestvina and M. Handel, Train tracks and automorphisms of freegroups, Ann. of Math. 135 (1992), 1–51.

[CL95] M. Cohen and M. Lustig, Very small group actions on R-trees and Dehntwist automorphisms, Topology 34 (1995), no. 3, 575–617.

[CM87] Marc Culler and JohnW. Morgan, Group actions on R-trees, Proc. LondonMath. Soc. (3) 55 (1987), no. 3, 571–604.

[CV86] M. Culler and K. Vogtmann, Moduli of graphs and automorphisms of freegroups, Invent. Math. 84 (1986), 571–604.

[FLP+79] A. Fathi, F. Laudenbach, V. Poenaru, et al., Travaux de Thurston sur lessurfaces, Asterisque, vol. 66–67, Societe Mathematique de France, 1979.

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[GJLL98] D. Gaboriau, A. Jaeger, G. Levitt, and M. Lustig, An index for countingfixed points of automorphisms of free groups, Duke Math. J. 93 (1998),no. 3, 425–452.

[Gui04] V. Guirardel, Core and intersection number for group actions on R-trees,preprint, arXiv:math.GR/0407206, 2004.

[HM04a] M. Handel and L. Mosher, The expansion factors of an outer automor-phism and its inverse, preprint, arXiv:math.GR/0410015, 2004.

[HM04b] M. Handel and L. Mosher, Axes in outer space: Irreducible automorphisms,in preparation, 2004.

[LL03] G. Levitt and M. Lustig, Irreducible automorphisms of Fn have North-South dynamics on compactified outer space, J. Inst. Math. Jussieu 2(2003), no. 1, 59–72.

[LM95] D. Lind and B. Marcus, An Introducion to Symbolic Dynamics and Chaos,Cambridge University Press, 1995.

[LP97] G. Levitt and F. Paulin, Geometric group actions on trees, Amer. J. Math.119 (1997), no. 1, 83–102.

[McC85] J. McCarthy, A “Tits-alternative” for subgroups of surface mapping classgroups, Trans. AMS 291 (1985), no. 2, 582–612.

[Vog02] K. Vogtmann, Automorphisms of free groups and outer space, Proceedingsof the Conference on Geometric and Combinatorial Group Theory, Part I(Haifa, 2000), vol. 94, 2002, pp. 1–31.

Michael Handel:

Department of Mathematics and Computer Science

Lehman College - CUNY

250 Bedford Park Boulevard W

Bronx, NY 10468

michael.handel@lehman.cuny.edu

Lee Mosher:

Department of Mathematics and Computer Science

Rutgers University at Newark

Newark, NJ 07102

mosher@andromeda.rutgers.edu

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