Introduction - uni-bonn.de2 URSULA HAMENSTADT¨ Let cv(Fn) be the unprojectivized Outer space of minimal free simplicial Fn- trees, with its boundary ∂cv(Fn) of all minimal very

THE BOUNDARY OF THE FREE FACTOR GRAPH

URSULA HAMENSTADT

Abstract. Define a tree T in the boundary of Outer Space to be arational

if T is indecomposable and if no measured lamination which annihilates T issupported in a free factor. We show that the Gromov boundary of the freefactor graph can be identified with the space of arational trees equipped with

a measure forgetful topology.

1. Introduction

A free factor in the free group Fn with n generators is a proper subgroup A ofFn so that Fn can be represented as a free product Fn = A ∗ B for some propersubgroup B of Fn.

The free factor graph is the graph FF whose vertices are conjugacy classes offree factors and where two free factors A,B are connected by an edge of length oneif and only if up to conjugation, either A < B or B < A. The free factor graph ishyperbolic in the sense of Gromov [BF11]. The outer automorphism group Out(Fn)of Fn acts on FF as a group of simplicial isometries.

The goal of this note is to determine the Gromov boundary of the free factorgraph.

Denote by ∂Fn the ideal boundary of Fn. This boundary is a compact Fn-space,and each element of Fn fixes precisely two points in ∂Fn. The set of all pairs offixed points of all elements in a given conjugacy class is a locally finite Fn-invariantsubset of the complement of the diagonal ∆ in ∂Fn × ∂Fn.

The space ML of measured laminations on Fn is the space of all locally finiteFn-invariant flip invariant Borel measures on the complement of the diagonal ∆in ∂Fn × ∂Fn which are limits in the weak∗-topology of atomic measures whosessupports consist of all pairs of fixed points of all elements in some primitive conju-gacy class. The projectivization PML of ML is a compact minimal Out(Fn)-space[KL07].

The Gromov boundary ∂A of a free factor A < Fn is naturally an A-invariantcompact subspace of ∂Fn. We say that a measured lamination µ is supported in

a free factor A of Fn if the support Supp(µ) of µ is contained in the Fn-orbit of∂A× ∂A−∆ ⊂ ∂Fn × ∂Fn −∆.

Date: November 18, 2012.AMS subject classification: 20M34

Partially supported by ERC grant 10160104.

1

2 URSULA HAMENSTADT

Let cv(Fn) be the unprojectivized Outer space of minimal free simplicial Fn-trees, with its boundary ∂cv(Fn) of all minimal very small Fn-trees [CL95, BF92].

Write cv(Fn) = cv(Fn) ∪ ∂cv(Fn). There is a continuous intersection form 〈, 〉 :

cv(Fn)×ML → [0,∞) [KL09]. If T ∈ cv(Fn) is free simplicial then 〈T, µ〉 > 0 forall µ.

An action of Fn on an R-tree T is called indecomposable if for any finite, non-degenerate arcs I, J ⊂ T , there are elements g1, . . . , gr ∈ Fn so that

J ⊂ g1I ∪ · · · ∪ grI

and such that giI ∩ gi+1I is non-degenerate for i ≤ r− 1 (Definition 1.17 of [G08]).By Lemma 1.18 of [G08], the Fn-orbits on an indecomposable tree are dense.

Denote by CV(Fn) the projectivization of cv(Fn), with its boundary ∂CV(Fn).

The union CV(Fn) = CV(Fn) ∪ ∂CV(Fn) is a compact space. The outer auto-

morphism group Out(Fn) of Fn acts on CV(Fn) as a group of homeomorphismspreserving the boundary ∂CV(Fn). An irreducible element of Out(Fn) with irre-ducible powers fixes precisely two points in ∂CV(Fn) [LL03].

Definition. A projective tree [T ] ∈ ∂CV(Fn) is arational if a representative T of[T ] has the following properties.

(1) T is indecomposable.(2) Let µ ∈ ML be any measured laminations with 〈T, µ〉 = 0. Then µ is not

supported in a proper free factor of Fn.

The space FT ⊂ ∂CV(Fn) of arational trees is Out(Fn)-invariant, but it isnot closed. It contains all fixed points of irreducible automorphisms of Fn withirreducible power.

If T ∈ cv(Fn) then denote by[T ] ∈ CV(Fn) the projectivization of T . Let ∼ bethe smallest equivalence relation on FT with the property that [S] ∼ [T ] if thereis a measured lamination µ ∈ ML such that 〈T, µ〉 = 〈S, µ〉 = 0.

We equip FT / ∼ with a measure forgetful topology. In this topology, a sequence[Ti] ⊂ FT / ∼ converges to [T ] ∈ FT / ∼ if and only if the following holds true. Let[µi] ⊂ PML be a sequence of projective measured laminations so that 〈Ti, µi〉 = 0for all i. By passing to a subsequence, assume that [µi] → [µ] ∈ PML; then〈T, µ〉 = 0. We refer to Section 6 for more details about this topology.

We show

Theorem. The Gromov boundary ∂FF of the free factor graph is the space FT / ∼of equivalence classes of arational trees with the measure forgetful topology.

The strategy of proof follows the strategy of Klarreich [K99] who determined theGromov boundary of the curve graph of a non-exceptional surface of finite type.The main difficulty lies in showing that arational trees are analogous to arationalmeasured laminations on a surface.

THE BOUNDARY OF THE FREE FACTOR GRAPH 3

As in [K99], we begin with describing in Section 2 an electrification of Outerspace which is quasi-isometric to the free factor graph. In Section 3 we establishsome properties of folding paths in cv(Fn) needed later on. Section 4 is devoted toa detailed analysis of trees in ∂cv(Fn) on which Fn does not act with dense orbits.In Section 5 we investigate arbitrary Fn-trees in ∂cv(Fn) which split as graphs ofactions. The proof of the theorem is completed in Section 6.

Acknowledgement: I am grateful to Vincent Guirardel and Gilbert Levitt foruseful discussions. In particular, I owe an argument used in the proof of Lemma 4.1to Vincent Guirardel. Thanks to Karen Vogtmann for help with references. Afterthis work was completed I learned that the results of this paper were a bit earlierobtained by Mladen Bestvina and Patrick Reynolds [BR12]. I am particularlyindebted to Martin Lustig for pointing out an error in an earlier version of thisnote.

2. The no gap metric

The goal of this section is to a construct a geometric model for the free factorgraph which is an Out(Fn)-invariant electrification of Outer space.

Denote by cv0(Fn) the Out(Fn)-invariant subspace of cv(Fn) of all free simplicialFn-trees with quotient of volume one. The restriction to cv0(Fn) of the canonicalprojection cv(Fn) → CV(Fn) is a homeomorphism.

Fix a (large) number k ≥ 2. For a tree T ∈ cv0(Fn) define a conjugacy classα in Fn to be primitive short if it is primitive and if it can be represented by apath of length at most k on T/Fn. Every T ∈ cv0(Fn) admits a primitive shortconjugacy class (see [FM11]). Call two trees T, T ′ ∈ cv0(Fn) tied if there is aprimitive conjugacy class α which is short for both T and T ′.

For T, T ′ ∈ cv0(Fn) let dng(T, T′) be the minimum of all numbers s ≥ 0 with the

following property. There is a sequence T = T0, . . . , Ts = T ′ ⊂ cv0(Fn) such thatfor all i the trees Ti, Ti+1 are tied. The following is immediate from the definitions.

Lemma 2.1. For sufficiently large k, dng is a metric on cv0(Fn).

Let

Υ : cv0(Fn) → FF

be a map which associates to a simplicial tree T with volume one quotient theconjugacy class of a free factor of rank one spanned by a primitive short conjugacyclass for T .

In the sequel we mean by a rose a marked metric rose with edges of equallength 1/n. Each rose R determines the conjugacy class of a free basis of Fn. Theuniversal covering T of R is the Cayley tree of this basis and is contained in cv0(Fn).If Fn = A ∗B is a free splitting of Fn we say that a rose R represents the splittingif R defines a basis a1, . . . , an of Fn so that for k = rk(A), the set a1, . . . , ak is abasis of A and ak+1, . . . , an is a basis of B. We have

4 URSULA HAMENSTADT

Proposition 2.2. The map Υ : (cv0(Fn), dng) → FF is a quasi-isometry.

Proof. Let T ∈ cv0(Fn) and let α be a primitive short conjugacy class for T . Sincethe volume of T/Fn is one and since T/Fn has at most 3g − 3 edges, there is anedge e of length at least 1/(3g − 3). By assumption, the conjugacy class α can berepresented by a loop γ on T/Fn of length at most k and hence γ passes through e atmost k(3g+3) times. By Lemma 3.2 of [BF11], this implies that the distance in FFbetween the conjugacy class 〈α〉 of the free factor spanned by α and the conjugacyclass of a free factor spanned by any other primitive short conjugacy class for Tis bounded by a number only depending on k. In particular, the distance between〈α〉 and Υ(T ) is uniformly bounded.

As a consequence, if T ′ ∈ cv0(Fn) is such that α is primitive short for T ′ thenthe distance between Υ(T ) and Υ(T ′) is uniformly bounded. This shows that themap Υ is coarsely Lipschitz with respect to the metric dng on cv0(Fn).

Define the ellipticity graph E as follows. Vertices of E are either conjugacy classesof free splittings of Fn or nontrivial cyclic words in Fn. Two vertices A ∗B and ware adjacent if w has a representative in A or B. The ellipticity graph is Out(Fn)-equivariantly quasi-isometric to the free factor graph [KL09]. Let dE be the distancein the ellipticity graph.

For each free splitting A ∗B of Fn choose a rose R(A ∗B) which represents this

free splitting. Let R(A ∗ B) be the universal covering of R(A ∗ B). It now sufficesto show the existence of a number ℓ > 1 such that for any two free splittings A ∗B,C ∗D we have

dng(R(A ∗B), R(C ∗D)) ≤ ℓdE(A ∗B,C ∗D).

To this end let A ∗B and C ∗D be free splittings of distance 2 in the ellipticitygraph. By Theorem 1.4 of [BK10], up to exchanging A,B and/or C,D and perhapsconjugating C ∗ D, there is some primitive element α ∈ A ∩ C. In particular, wehave

Fn = 〈α〉 ∗A′ ∗B = 〈α〉 ∗ C ′ ∗D

for some free factor A′ of A and C ′ of C.

Choose roses R,R′ realizing these two free splittings (up to conjugation), withuniversal coverings T, T ′. For appropriate choices of R,R′ there is a petal in R(A ∗B), R(C ∗D) which defines a conjugacy class in Fn also defined by a petal in R,R′.

Then T and R(A ∗B) are tied, and the same holds true for T ′ and R(C ∗D). Thus

the no-gap distance between T, R(A ∗B) and between T ′, R(C ∗D) is at most one.

The minimal length of a loop in R,R′ representing α is at most one and hencethe no-gap distance between T, T ′ equals at most one as well. Together we concludethat the no-gap distance between R(A ∗ B) and R(C ∗ D) is at most three. Theproposition follows.


3. Folding paths

A morphism between Fn-trees S, T is an equivariant map ϕ : S → T such thatevery segment of S can be subdivided into finitely many subintervals on which ϕ isan isometric embedding.

The following (well known) construction is taken from Section 2 of [BF11]. Letfor the moment U be an arbitrary Fn-tree. A direction at a point x ∈ U is a germof non-degenerate segments [x, y] with y 6= x. At each interior point of an edge ofU there are exactly two directions. A collection of directions at x is called a gate

at x. A turn at x is an unordered pair of distinct directions at x. It is called illegal

if the directions belong to the same gate, and it is called legal otherwise. A train

track structure on U is an Fn-invariant family of gates at the points of U so thatat each x ∈ U there are at least two gates.

A morphism ϕ : S → T determines a collection of gates as follows. Define aturn in S to be illegal if it is given by two directions which are identified by themorphism ϕ. Otherwise the turn is called legal. Two directions d, d′ at the samepoint belong to the same gate if either d = d′ or if the turn d, d′ is illegal. If thesegates determine a train track structure on S and if moreover there is a train trackstructure on T so that ϕ is an isometric embedding on each edge and legal turnsare sent to legal turns, then ϕ is called a train track map (see p.7 of [BF11]).

Recall that there is a natural bijection between conjugacy classes of free bases ofFn and roses (=marked metric roses with n petals). Define the standard simplex of

a free basis of Fn to consist of all simplicial trees U ∈ cv0(Fn) which are universalcoverings of graphs of volume one obtained from the rose R corresponding to thebasis by changing the lengths of the edges. Note that we allow that U is contained inthe boundary of unprojectivized Outer space, i.e. that the rank of the fundamentalgroup of the graph U/Fn is strictly smaller than n.

The construction in the following lemma is discussed in detail in Section 2 of[BF11]. We also refer to this paper for references to earlier works where this con-struction is introduced.

Lemma 3.1. For every [T ] ∈ CV(Fn) and every standard simplex ∆ there is a tree

U ∈ ∆ and a train track map ϕ : U → T where T is some representative of [T ].

Proof. In the case that [T ] ∈ CV(Fn) is free simplicial a detailed argument is givenin the proof of Proposition 2.5 of [BF11] (see also [FM11]). A limiting argumentthen yields the result for trees [T ] ∈ ∂CV(Fn).

Namely, let [Ti] ⊂ CV(Fn) be a sequence of free simplicial projective trees which

converge in CV(Fn) to a projective tree [T ]. Let Σ be the set of all trees U ∈ cv(Fn)such that the minimum over all trees S ∈ ∆ of the optimal Lipschitz constant forequivariant maps S → U equals one. By Proposition 2.5 of [BF11], for each i thereis a point Si ∈ ∆, a representative Ti ∈ Σ of [Ti] and a train track map fi : Si → Tiof Lipschitz constant one.

6 URSULA HAMENSTADT

The set Σ is compact with respect to the equivariant Gromov Hausdorff topology[P88]. For each i the graph Ai of fi is a closed Fn-invariant subset of Si × Ti.By compactness, we may assume that the sequence Ai converges to a closed Fn-invariant subset A of S × T where S ∈ ∆ and T ∈ Σ is a representative of [T ].

By continuity, for (x1, y1), (x2, y2) ∈ A we have dT (y1, y2) ≤ dS(x1, x2). Inparticular, for each x ∈ S there is a unique point f(x) ∈ T so that (x, f(x)) ∈ A,and the assignment x→ f(x) is an equivariant map S → T with optimal Lipschitzconstant one. This map is an isometry on edges, and there are at least two gatesat each preimage of the unique vertex of S/Fn. In other words, f has the requiredproperties.

Let S ∈ cv(Fn) and let f : S → T be a train track map. By the definitionof a train track map, f embeds every edge. Let ǫ > 0 be half of the smallestlength of an edge of S. Let e, e′ be edges with the same initial vertex v whichdefine an illegal turn. Assume that e, e′ are parametrized by arc length on compactintervals [0, a], [0, a′] where e(0) = e′(0) = v. Then there is some t ∈ (0, ǫ] so thatfe[0, t] = fe′[0, t]. For s ∈ [0, t] let Ss be the quotient of S by the equivalencerelation ∼s which is defined by u ∼s v if and only if u = ge(r) and v = ge′(r) forsome r ≤ s and some g ∈ Fn. The tree Ss is called a fold of S obtained by foldingthe illegal turn defined by e, e′ (once again, compare the discussion in Section 2 of[BF11]). Note that for s > 0 the volume of the graph Ss/Fn is strictly smaller thanthe volume of S/Fn. There also is an obvious notion of a maximal fold at the turndefined by e, e′.

Using the terminology of the previous paragraph, the assignment s → Ss (s ∈[0, t]) is a path in cv(Fn) through S0 = S which is called a folding path. Thesemigroup property holds for folding paths. For each s ∈ [0, t] the train track mapf : S → T decomposes as f = fs ϕs where fs : Ss → T is a train track map andϕs is a train track map for the train track structures on S, Ss defined by f andfs. We refer to [HM11] and to Section 2 of [BF11] for details of this construction.We insist that we view the initial train track map f : S → T as part of the datadefining a folding path (this prevents going backwards along the path).

Repeat this construction with St and a perhaps different pair of edges. Thepath constructed in this way by successive foldings terminates if T is free simplicial(Proposition 2.2 of [BF11]).

In case we rescale all trees along the path to have volume one quotient andrescale the endpoint tree T accordingly it makes sense to fold with unit speed eachof the illegal turns at once. Using Proposition 2.2 and Proposition 2.5 of [BF11](compare [BF11] for references), a path constructed in this way from a train trackmap f : S → T is unique and will be called a Skora path in the sequel. Notehowever that the path depends on the train track map f : S → T . If T ∈ cv(Fn)then this path has finite length, otherwise its length may be infinite.

We summarize the discussion as follows (see also Proposition 2.5 of [BF11]).

Lemma 3.2. For every standard simplex ∆ and every tree [T ] ∈ CV(Fn) there is

a Skora path connecting some point in ∆ with a representative T of [T ].


Proof. By Lemma 3.1, there is a representative T of [T ], a tree U ∈ ∆ and atrain track map f : U → T . This train track map then determines a uniqueSkora path (xt) issuing from U . By construction, the projectivizations [xt] of

the trees xt converge as t → ∞ in CV(Fn) to the projectivization [T ] of T (see[FM11, BF11]).

In the sequel we will use volume renormalization to define a Skora path. However,most of the time we consider unnormalized Skora paths, i.e. we scale the trees backin such a way that the train track maps along the path are edge isometries onto afixed endpoint tree T .

For a number L > 1, an L-quasi-geodesic in FF is a path ρ : J ⊂ R → FF suchthat

|s− t|/L− L ≤ d(ρ(s), ρ(t)) ≤ L|s− t|+ L

for all s, t ∈ J . The path ρ is called an unparametrized L-quasi-geodesic if there isa homeomorphism ψ : I → J so that ρ ψ : I → FF is an L-quasi-geodesic.

Extend the map Υ : cv0(Fn) → FF defined in Section 2 to arbitrary treesS ∈ cv(Fn) by requiring that Υ(aT ) = Υ(T ) for all T ∈ cv0(Fn) and all a > 0.There also is a natural extension of Υ to the subset of ∂cv(Fn) which consists of allminimal very small simplicial Fn-trees with trivial edge stabilizers. In particular, Υis defined on any standard simplex ∆, and the image of ∆ under Υ has uniformlybounded diameter independent of ∆. The following result is Corollary 5.5 of [BF11].

Proposition 3.3. There is a number L > 1 such that the image under Υ of a

folding path is an unparametrized L-quasi-geodesic in FF .

For a number c > 0, we say that a path α : [0,∞) → FF is a c-fellow traveler

of a path β if there is a nondecreasing function τ : [0,∞) → [0,∞) such that forall t ≥ 0 we have d(α(t), β(τ(t))) ≤ c. Note that we do not require here that thefunction τ has unbounded image.

Lemma 3.4. There is a number c > 0 with the following property. Let T ∈∂cv(Fn), let U ∈ cv(Fn) be a point in a standard simplex and let f : U → T be

a train track map. If (wt) ⊂ cv(Fn) is the Skora path defined by f and if (yt) is

any folding path defined by f then the path Υ(yt) is a c-fellow traveller of the path

Υ(wt).

Proof. For simplicity of notation, in this proof we use a different parametrizationof a Skora path. Namely, we do not renormalize the volume of the quotient graphsalong the path, but we always identify edges with speed one. We call such a patha reparametrized Skora path.

Let (ys)s≥0 be a folding path defined by a train track map f : y0 → T . Again weidentify segments with speed one without renormalization. Let moreover (ws)s≥0

be the reparametrized Skora path defined by the same train track map. Since wedo not renormalize volume, for each s there is a train track map gs : ws → T . Weclaim that for all s there is a train track map ϕs : ys → ws whose composition withgs is a train track map gs ϕs : ys → T .

8 URSULA HAMENSTADT

To this end let A ⊂ [0,∞) be the set of all numbers s with this property. Thenclearly 0 ∈ A, moreover A is closed. It now suffices to show that A is open.

Thus let s ∈ A and let ϕs : ys → ws be a train track map whose existence isassumed by the definition of A. For δ > 0, the composition of ϕs with the naturaltrain track map ws → ws+δ is a train track map Φδ : ys → ws+δ, for the given traintrack structures on ys, ws+δ, and

gs+δ Φδ : ys → T

is a train track map.

Let (xs+ǫ) (ǫ ∈ [0, δ]) be the initial segment of length δ of the reparametrizedSkora path which starts at ys and is determined by the train track map Φδ : ys →ws+δ. By Lemma 2.3 of [BF11], for all ǫ ≤ δ, the tree xs+ǫ is contained in the Skorapath connecting ys to T which is determined by the train track map gsϕs : ys → T .In particular, xs+ǫ admits a natural train track structure so that the morphisms

Ξǫ : ys → xs+ǫ and Ψǫ : xs+ǫ → ws+ǫ

are train track maps for this structure.

Let ǫ0 > 0 be smaller than half the smallest length of an edge of ys. Forsufficiently small ǫ ≤ ǫ0, say for all ǫ ≤ ǫ1, the tree xs+ǫ can be obtained from ys byidentifying all initial segments of edges defining an illegal turn on a length of ǫ. Onthe other hand, by perhaps decreasing ǫ1, we may assume that for ǫ ≤ ǫ1 the treeys+ǫ is obtained from ys by folding intial segments of some of the edges defining anillegal turn in ys. This implies that there is a train track map Θǫ : ys+ǫ → xs+ǫ

whose composition with the train track map Ψǫ : xs+ǫ → ws+ǫ is a train track mapΨǫ Θǫ : ys+ǫ → ws+ǫ. This map in turn can be composed with gs+ǫ to a traintrack map ys+ǫ → T . This shows that A is open.

As a consequence, for each s a folding path connecting y0 to ys can be composedwith a folding path connecting ys to ws, and this defines a folding path connectingy0 to ws. Remember here that we required a train track map to be part of the datadefining a folding path.

The concatentation of this folding path with the path (wt)t≥s is a folding path(zsu) connecting y0 to T . Now the image under Υ of a folding path is a uniformunparametrized quasi-geodesic in FF . Since the paths (wt) and (zsu) issue fromthe same point and coincide evenutally, by hyperbolicity of FF the path Υ(zsu)is a c-fellow traveller of Υ(wt) for a number c > 0 only depending on FF . Sinces > 0 was arbitrary we conclude that Υ(yt) is a uniform fellow traveller of the pathΥ(wt).

Since (ys) was an arbitrary folding path defined by the train track map f : y0 →T , this shows the lemma.

Define an equivalence relation ≡ on FF ∪ ∂FF by x ≡ y if and only if eitherx = y or if x, y ∈ FF .

Lemma 3.2 and Proposition 3.3 imply that for every standard simplex ∆ thereis a natural coarsely defined map ϕ∆ : ∂CV(Fn) → FF ∪ ∂FF/ ≡. The map


ϕ∆ associates to a projective tree [T ] ∈ ∂CV(Fn) the equivalence class of theendpoint of the image under Υ of a Skora path connecting a starting point in ∆ toa representative of [T ].

Namely, let (xt) be a Skora path connecting a point in ∆ to T . If the pathΥ(xt) has finite diameter then define ϕ∆([T ]) to be the class of a point in FF . Ifthe path Υ(xt) has infinite diameter then ϕ∆([T ]) is the unique endpoint in theGromov boundary ∂FF of FF of the unparametrized L-quasi-geodesic Υ(xt).

Note that a priori, this map depends on choices since a Skora path connecting atree S in ∆ to a representative T of [T ] depends on the choice of a train track mapS → T . The next lemma shows that the map ϕ∆ does not depend on any choicesmade.

Lemma 3.5. The map ϕ∆ does not depend on the choice of the Skora path or on

the choice of ∆. Moreover it is Out(Fn)-equivariant.

Proof. Let (xt)t≥0 be a Skora path which connects a point x0 ∈ ∆ to a tree T ∈∂cv(Fn). Choose another (not necessarily different) standard simplex Λ and anarbitrary train track map f : S → aT where S ∈ Λ and a > 0. Since the quotientS/Fn has a single vertex v, by equivariance and the definition of a train track map,f is uniquely determined by f(v) where v is a preimage of v in S.

Now xt → T in cv(Fn) (t → ∞) and hence by the definition of the equivariantGromov Hausdorff topology [P89] and the discussion in the proof of Lemma 3.1,there is a sequence ti → ∞, a sequence Si ⊂ Λ of points in the standard simplexΛ, a sequence ai → a and a sequence of train track maps fi : Si → aixti whichconverge as i→ ∞ to the map f in the following sense: The graphs of fi in Si×aixticonverge in the equivariant Gromov Hausdorff topology to the graph of f .

Namely, let g1, . . . , gn be a free basis of Fn which determines the simplexΛ and let K be the compact subtree of T which is the convex hull of the pointsf(v), gjf(v), g

−1j f(v) (j = 1, . . . , n). By definition of the equivariant Gromov Haus-

dorff topology, for each t there is a compact subtree Kt of xt which is the convexhull of a point vt ∈ xt and the points gjvt, g

−1j vt so that the trees Kt converge to

K in the usual Gromov Hausdorff topology (see [P89] for details). Since for eachU ∈ Λ the quotient tree U/Fn has a single vertex, for each t there there is a treeSt ∈ Λ, there is a number at > 0 and an equivariant train track map St → atxtwhich maps a preimage of a vertex of St/Fn to vt. By [P88], a subsequence of thissequence of train track maps has the requested properties.

The subspace of cv(Fn) of all trees which have one-sided Lipschitz distance onefrom a tree in Λ is compact. For each i connect Si to aixti by an arbitrary Skorapath (yis). Now normalized Skora paths are geodesics for the one-sided Lipschitz

metric on the locally compact space cv0(Fn) [FM11] and hence we can apply to thesepaths the Arzela Ascoli theorem. As a consequence, up to passing to a subsequencethe paths (yis) converge as i → ∞ to a Skora path (ys) connecting S to aT . Theimage under Υ of this family of paths is a family of unparametrized quasi-geodesicsin FF .

10 URSULA HAMENSTADT

Recall that the diameter of Υ(Λ) ⊂ FF is bounded independent of Λ. Let b ≥ 0be such that Υ(xb) is a coarsely well defined shortest distance projection of Υ(Λ)into the unparametrized quasi-geodesic Υ(xt). By hyperbolicity of FF , for everyu > b and every v > u, the image under Υ of any Skora path connecting a pointin Λ to xv passes through a uniformly bounded neighborhood of Υ(xu). Thus forall u > b and all i such that ti > u, Υ(yis) passes through a uniformly boundedneighborhood of Υ(xu) and hence the same holds true for Υ(ys). This shows thatthere is a number R > 0 only depending on Λ,∆ such that Υ(ys) is an R-fellowtraveller of Υ(xt). By symmetry, we conclude that indeed the map ϕ∆ coarselydoes not depend on the choice of ∆ or on the choices of the Skora paths. As aconsequence, the map ϕ∆ is moreover coarsely Out(Fn) equivariant.

4. Trees in ∂CV(Fn) without dense orbits

By the results in [CL95, BF92], the boundary ∂cv(Fn) of unprojectivized Outerspace cv(Fn) consists of minimal very small actions of Fn on R-trees. This meansthat a point in ∂cv(Fn) is a minimal Fn-tree with the following properties.

(1) Edge stabilizers are cyclic.(2) If gn stabilizes an edge e for some n ≥ 1 then so does g.(3) Fix(g) contains no tripod for g 6= 1.

A tree T ∈ ∂cv(Fn) decomposes canonically into two disjoint Fn-invariant sub-sets Td and Tc. Here Td is the set of all points p such that the orbit Fnp is discrete,and Tc = T −Td. The set Tc ⊂ T is closed. Each of its connnected components is asubtree T ′ of T . The stabilizer of T ′ acts on T ′ with dense orbits. We have Td = ∅if and only if the group Fn acts on T with dense orbits. This property is invariantunder scaling and hence it is defined for projective trees.

Let Θ be the equivalence class of FF in FF ∪ ∂FF/ ≡. Using the notationsfrom Section 3, the goal of this section is to show that ϕ∆([T ]) = Θ for every treeT with Td 6= ∅.

Thus let T ∈ ∂cv(Fn) be a very small Fn-tree with Td 6= ∅. The quotient T/Fn

admits a natural pseudo-metric. Let T/Fn be the associated metric space.

Since T is very small, by Theorem 1 of [L94] the space T/Fn is a finite graph.Edges correspond to orbits of the action of G on π0(T/B) where B ⊂ T is the set

of branch points of T . The graph T/Fn defines a graph of groups decompositionfor Fn, with at most cyclic edge groups.

The proof of the following observation is inspired by an argument which wasshown to me by Vincent Guirardel.

Proposition 4.1. If [T ] ∈ ∂CV(Fn) is such that Td 6= ∅ then ϕ∆([T ]) = Θ.


Proof. Let [T ] ∈ ∂CV(Fn) be such that Td 6= ∅. Let (xt) be an unnormalized Skorapath connecting a point x0 in a standard simplex ∆ to a representative T of [T ].

Let e be an edge of T/Fn.

For t ≥ 0 let ft : xt → T be the morphism defined by the Skora path. Themap ft is equivariant and surjective. Moreover, for all s ≤ t there is a morphismgst : xs → xt such that fs = ft gst.

Let a > 0 be the smallest length of an edge in T/Fn and let c ≥ a be the volume

of T/Fn. The volume of xt/Fn is a decreasing function in t which converges ast → ∞ to some b ≥ c. Thus there is a number t0 > 0 such that the volume ofxt0/Fn is smaller than b+ a/8.

Let e0 be an edge of T . The length of e0 is at least a. By equivariance and thefact that for each t the map ft is an edge isometry, there is a non-degenerate subarcc0 of e0 of length at least a/2 (the central subarc of length a/2) such that for eacht ≥ t0 the preimage of c0 under ft consists of k(t) ≥ 1 pairwise disjoint subarcs ofedges of xt0 (i.e. the preimage does not contain any vertex). Namely, since ft is anedge isometry, otherwise there are two segments in xt of length at least a/4 whichare identified by the Skora path connecting xt to T . By equivariance, this impliesthat there is some u > t such that the volume of xu/Fn does not exceed the volumeof xt/Fn minus a/4. This violates the assumption on the volume of xt0/Fn.

We claim that the number k(t) of preimages of c0 in xt does not depend ont ≥ t0. Namely, by equivariance and the definition of a Skora path, otherwise thereis some t > 0 and a vertex of xt which is mapped by ft into c0. By the discussionin the previous paragraph, this is impossible.

Write k = k(t0). For each u ≥ t0 the preimage of c0 in xu consists of exactlyk segments. Then for each u ≥ t0, the morphism gt0u maps the k components ofthe preimage of c0 in xt0 isometrically onto the k components of the preimage ofc0 in xu. By equivariance, this implies that there is an edge ht0 of xt0/Fn andthere is a subsegment ρt0 of ht0 of length a/2 which is mapped by the quotient mapgt0u : xt0/Fn → xu/Fn of the train track map gt0u isometrically onto a subsegmentρu of an edge hu in xu/Fn, and g

−1t0u

(ρu) = ρ0.

As a consequence, a loop in xt0/Fn which passes through the edge ht0 at mostm ≥ 0 times is mapped by gt0u to a loop in xu/Fn which passes through hu atmost m ≥ 0 times. By Lemma 3.2 of [BF11], this implies that the distance inFF between Υ(xt0) and Υ(xu) is uniformly bounded, independent of u ≥ t0. As aconsequence, we have ϕ∆([T ]) = Θ as claimed.

5. Trees which split as graph of actions

The goal of this section is to investigate the structure of those trees in ∂CV(Fn)with dense orbits which have a structure similar to the structure of trees T withTd 6= ∅. The description of such trees is as follows [G08, L94].

Definition 5.1. A graph of actions consists of


(1) a simplicial tree S, called the skeleton, equipped with an action of Fn

(2) for each vertex v of S an R-tree Yv, called a vertex tree, and(3) for each oriented edge e of S with terminal vertex v a point pv ∈ Yv, called

an attaching point.

It is required that the projection Yv → pv is equivariant and that for g ∈ Fn onehas gpe = pge.

Associated to a graph of actions G is a canonical action of Fn on an R-tree TGwhich is called the dual of the graph of actions [L94]. Define a pseudo-metric d on∐

v∈V (S) Yv as follows. If x ∈ Yv0, y ∈ Yvk

let e1 . . . ek be the reduced edge-path

from v0 to vk in S and define

d(x, y) = dYv1(x, pe1) + · · ·+ dYvk

(pek , y).

Making this pseudo-metric Hausdorff gives an R-tree TG .

If T is an Fn-tree and if there is an equivariant isometry T → TG to the dual ofa graph of actions then we say that T splits as a graph of actions. In particular,every tree T with Td 6= ∅ splits as a graph of actions, but there are trees T whichsplit as a graph of actions with Td = ∅.

We also say that the projectivization [T ] of an Fn-tree T splits as a graph ofactions if T splits as a graph of actions.

A transverse family for an Fn-tree S with dense orbits is an Fn-invariant familyYv of non-degenerate subtrees Yv ⊂ T with the property that if Yv 6= Yv′ thenYv∩Yv′ contains at most one point. The transverse family is a transverse covering ifany finite segment I ⊂ T is contained in a finite union Yv1

∪· · ·∪Yvrof components

from the family. By Lemma 1.5 of [G08], T admits a transverse covering if andonly if T splits as a graph of actions.

An alignment preserving map between two Fn-trees T, T′ ∈ cv(Fn) is an equi-

variant map f : T → T ′ with the property that x ∈ [y, z] implies f(x) ∈ [f(y), f(z)].An equivariant map f is alignment preserving if and only if the preimage of ev-ery point in T ′ is convex [G00]. The map f is then continuous on segments. Analignment preserving morphism is an equivariant isometry.

If T ∈ ∂cv(Fn) splits as a graph of actions then there is a minimal Fn-tree Uwith Ud 6= ∅, and there is an alignment preserving one-Lipschitz map U → T whichrestricts to an equivariant isometry on the preimage of each vertex tree of T . Wecall U an extension of T . If Td 6= ∅ then we can choose U = T . Otherwise thetree U can be obtained by inserting the skeleton of T , however U is by no meansunique.

The next lemma shows that U can be chosen to be very small. Note howeverthat this is not true for every such tree U (see [R12] for an example).

Lemma 5.2. If [T ] ∈ ∂CV (Fn) has dense orbits and splits as a graph of actions

then there is a tree T ′ ∈ ∂cv(Fn) with T′d 6= ∅, and there is a one-Lipschitz alignment

preserving map ρ : T ′ → T .


Proof. Assume that T ∈ ∂cv(Fn) has dense orbits and splits as a graph of actions.Choose an extension S of T . Then there is a one-Lipschitz alignment preservingmap f : S → T whose restriction to the preimage of each of the vertex trees ofthe transverse covering defining the graph of actions is an isometry. The map fcollapses each edge of S to a point.

Fix an edge e in S/Fn and let V be the tree obtained from S by equivariantlycollapsing those edges of S to points which do not project to e. Then V is a minimalFn-tree with Vd 6= ∅. Moreover, there is an equivariant surjective one-Lipschitzalignment preserving map

ξ : V → T

which collapses the edges of V to points. More precisely, the image under ξ of thethe set of edges of V is a countable Fn-invariant collection A of points in T . Thetree V defines the structure of a graph of actions with dual tree T . We denote byYv (v ∈ A) the corresponding transverse covering of T .

Since T is very small, the tree V is very small if the stabilizer of an edge in V isat most maximal cyclic. In this case we are done.

Let U ∈ cv(Fn) be a point in a standard simplex ∆ and let

ϕ : U → T

be a train track map. Then ϕ is determined by the image of a vertex v ∈ U . Namely,let e1, . . . , en be a free basis of Fn whose conjugacy class defines the standardsimplex ∆ and let v be a vertex of U . By equivariance and the definition of amorphism, ϕ is determined by the point ϕ(v) ∈ T , the points ϕ(eiv) (i = 1, . . . , n)and the requirement that the restriction of ϕ to each edge of U is an isometricembedding.

We use the map ϕ to construct a simplicial tree U ′, a one-Lipschitz alignmentpreserving map α : U ′ → U and a morphism β : U ′ → V as follows.

Assume first that ϕ(v) 6∈ A. Then the points ϕ(v), ϕ(eiv) have unique preimagesβ(v) = ξ−1(ϕ(v)), β(eiv) = ξ−1(ϕ(eiv)) in V . There is an Fn-tree U

′ ∈ cv(Fn)which can be obtained from U by equivariantly rescaling the edges of U , and thereis a train track map

β : U ′ → V

defined by the points β(v), β(eiv) (here we use the same notation v for the vertex inU and U ′). Moreover, there is a one-Lipschitz alignment preserving map α : U ′ → Usuch that

ξ β = ϕ α.

The map α contracts some closed subintervals of edges of U ′ to points.

Similarly, if ϕ(v) ∈ A then ϕ(eiv) ∈ A for all i. Then ξ−1(ϕ(v)) is an edge inV with a well defined midpoint β(v). Construct the tree U ′ and the map β fromthese data as before by equivariance.

The map ϕ determines an unnormalized Skora path (xt) connecting U = x0 toT . Let ϕt : xt → T be the corresponding family of train track maps. For t > s wehave ϕs = ϕt gst for a train track map gst : xs → xt. The above recipe (lifting


those vertices of xt which are not mapped into A to the corresponding points in V ,lifting the vertices which are mapped into A to the midpoints of the correspondingedges and equivariantly adjusting lengths) can be used to construct for each t anFn-tree yt, a morphism βt : yt → V and a one-Lipschitz alignment preserving mapαt : yt → xt such that

ξ βt = ϕt αt.

For each t, the tree yt is very small simplicial.

Let again v ∈ x0 be a vertex, let P be any finite subset of Fn and consider theconvex hulls of the images of v by the elements of P . Then K is a compact subtreeof x0. The image ϕ(K) of K in T is a compact subtree of T . By the definition ofa graph of actions, this subtree intersects the set A in only finitely many points.

Let m ≥ 0 be this number of points, let n ≥ 1 be the cardinality of P and letk ≥ 0 be the total length of K, i.e. the sum of the length of all edges contained inK. If we assume without loss of generality that the length of the edges of V equalsone then the total length of the subtree of U ′ constructed from the set P and thepreimage of the vertex v does not exceed k +mn.

Now for each t the total length of the subtree of xt which is the convex hullof the points g0tv, hg0tv (h ∈ P ) does not exceed k, and hence as in the previousparagraph, the total length of the subtree Kt of yt defined by P and the preimage ofg0tv is not bigger than k+mn. As a consequence, for each ǫ > 0 there is a coveringof these trees by balls of radius ǫ whose cardinality does not depend on t. Thus bythe main theorem of [P88], there is a sequence ti → ∞ such that the sequence (yti)converges in the equivariant Gromov Hausdorff topology to an Fn-tree y. The treey is very small [CL95, BF92], and yd 6= ∅.

Since xti → T and since for each t there is a one-Lipschitz alignment preservingmap αt : yt → xt, by the definition of the equivariant Gromov Hausdorff topologyand by passing to a subsequence we may assume that there is a one-Lipschitzalignment preserving map

ρ : y → T.

This is what we wanted to show.

The proof of the next proposition is similar to the proof of Lemma 5.2.

Lemma 5.3. Let T, T ′ ∈ ∂cv(Fn) and assume that there is a one-Lipschitz align-

ment preserving map ρ : T → T ′. Then ϕ∆([T |) = ϕ∆([T′]).

Proof. Let S be a free simplicial Fn-tree and let ϕ : S → T be a train track map.Assume that there is a one-Lipschitz alignment preserving map ρ : T → T ′. Thenthe map ρ ϕ : S → T ′ is equivariant and one-Lipschitz. In particular, there is anFn-tree S

′ which can be obtained from S by decreasing the lengths of some edges ofS, there is a one-Lipschitz alignment preserving map α : S → S′ and a morphismϕ′ : S′ → T ′ such that

ϕ′ α = ρ ϕ.

Here we allow that the map α collapses some edges to points.


Let a, b ⊂ S be subsegments of edges incident on the same vertex p which areidentified by the map ϕ. Then the segments a, b are also identified by ρ ϕ. Thismeans the following. Let U be the simplicial tree obtained by folding the segmentsa, b. Then there is a morphism ψ : U → T . Moreover, there is a simplicial treeU ′ which can be obtained from S′ by a (perhaps trivial) fold, and there is a one-Lipschitz alignment preserving map β : U → U ′ such that

ψ′ β = ρ ψ

for a morphism ψ′ : U ′ → T ′.

This discussion shows the following. Let (xt) be a folding path connecting S toT . By this we mean that only a single fold is performed at the time, and (xt) → Tin the equivariant Gromov Hausdorff topology. For each t there is a train trackmap ht : xt → T . There is a (suitably parametrized) folding path (yt) connectingS′ to T ′, for each t there is a one-Lipschitz alignment preserving map ζt : xt → yt,and there is a morphism h′t : yt → T ′ such that the diagram

xtht−−−−→ T

ζt

y

ρ

y

yth′

t−−−−→ T ′

commutes. Moreover, as t→ ∞, yt converges in the equivariant Gromov Hausdorfftopology to T ′.

For each t, the distance in FF between Υ(xt) and Υ(yt) is uniformly bounded.Since the assignments t→ Υ(xt) and t→ Υ(yt) are uniform unparametrized quasi-geodesics, the lemma now follows from Lemma 3.4.

As an immediate corollary of Lemma 5.2 and Lemma 5.3 we obtain

Corollary 5.4. If [T ] splits as a graph of actions then ϕ∆([T ]) = Θ.

From now on we only consider trees T ∈ ∂cv(Fn) (or projective trees [T ] ∈∂CV(Fn) with dense orbits. For simplicity we call such a tree dense.

The following definition is due to Paulin (see [G00]).

Definition 5.5. A length measure µ on T is an Fn-invariant collection

µ = µII⊂T

of locally finite Borel measures on the finite arcs I ⊂ T ; it is required that for J ⊂ Iwe have µJ = (µI)|J .

The Lebesgue measure λ defining the metric on T is an example of a lengthmeasure on T with full support.

Denote by M0(T ) the set of all non-atomic length measures on T . By Corollary5.4 of [G00],M0(T ) is a finite dimensional convex set which is projectively compact.Up to homothety, there are at most 3n − 4 non-atomic ergodic length measures.Each non-atomic length measure µ ∈ M0(T ) defines an Fn-tree Tµ ∈ ∂cv(Fn) as


follows [G00]. Define a pseudo-metric dµ on T by dµ(x, y) = µ([x, y]). Making thispseudo-metric Hausdorff gives an R-tree Tµ.

By Lemma 10.2 of [R10], if T admits an invariant atomic measure then T splitsas a graph of actions. The same holds true if T admits two invariant non-atomicmeasures whose supports are non-degenerate and distinct (Lemma 12.1 of [R10]).Thus if either T admits an invariant atomic measure or two invariant measures withdistinct non-degenerate support then ϕ∆([T ]) = Θ.

Following [R10], if T, T ′ ∈ cv(Fn) and if there is an alignment preserving mapf : T → T ′ then we say that T ′ is a projection of T . By the discussion in [G00], ifµ′ is a non-atomic length measure on T ′ then there is a length measure µ on T suchthat f∗µ = µ′. This means that for every segment I ⊂ T we have µ(I) = µ′(fI).We use Lemma 5.3 to show

Corollary 5.6. Let T, T ′ ∈ ∂cv(Fn) and let f : T → T ′ be alignment preserving.

Then ϕ∆([T ]) = ϕ∆([T′]).

Proof. Let T, T ′ ∈ ∂cv(Fn) and assume that there is an alignment preserving mapf : T → T ′. Then there are length measures ξ ∈ M0(T ), ξ

′ ∈ M0(T′), and there is

a one-Lipschitz map f : Tξ → Tξ′ . Lemma 5.3 shows that ϕ∆([Tξ]) = ϕ∆([Tξ′ ]).

Thus it suffices to show the following. Let µ ∈ M0(T ) be arbitrary; thenϕ∆([Tµ]) = ϕ∆([T ]).

To this end let ν be a length measure which is contained in the interior of theconvex polyhedron M0(T ). Let ζ be one of the vertices of M0(T ); this is an ergodicmeasure in M0(T ). Up to rescaling, there is a one-Lipschitz alignment preservingmap Tν → Tζ . Lemma 5.3 shows that ϕ∆([Tν ]) = ϕ∆([Tζ ]).

Now if ξ ∈ M0(T ) is arbitrary then there is an ergodic measure β ∈ M0(T ),and there is a one-Lipschitz alignment preserving map Tξ → Tβ . Using once moreLemma 5.3, we deduce that ϕ∆([Tξ]) = ϕ∆([Tβ ]) = ϕ∆([Tν ]). Since the Lebesguemeasure on T defines a point inM0(T ) this completes the proof of the corollary.

As an immediate consequence of Lemma 4.1 and Corollary 5.6, we have

Corollary 5.7. If the tree [T ] admits an alignment preserving map onto a tree

which splits as a graph of actions then ϕ∆([T ]) = Θ.

Recall from the introduction the definition of an indecomposable tree. Thefollowing result is Corollary 11.2 of [R10].

Proposition 5.8. Let T ∈ cv(Fn) have dense orbits, and assume that T is nei-

ther indecomposable nor splits as a graph of actions. Then there is an alignment

preserving map f : T → T ′ such that

(1) either T ′ is indecomposable or T ′ splits as a graph of actions,

(2) the image under f of the zero lamination of T is contained in the zero

lamination of T ′.


Proposition 5.8, Proposition 4.1 and Corollary 5.7 immediately imply

Corollary 5.9. Let [T ] ∈ ∂CV(Fn) be such that ϕ∆([T ]) ∈ ∂FF . Then T admits

an alignment preserving map onto an indecomposable tree.

Proof. By Proposition 4.1, if ϕ∆([T ]) ∈ ∂FF then T has dense orbits. The corollarynow follows from Proposition 5.8 and Corollary 5.7.

6. Arational trees

In this section we complete the proof of the theorem from the introduction.

The zero lamination L2(T ) of an R-tree T is the closed Fn-invariant subset of∂Fn×∂Fn−∆ which is the set of all accumulation points of pairs of fixed points ofany family of conjugacy classes with translation length on T that tends to 0. Thezero lamination of a tree T ∈ cv(Fn) is empty. For T ∈ ∂cv(Fn), it only depends onthe projective class [T ] ∈ ∂CV(Fn) of T . We refer to [CHL07] for more informationon the zero lamination of a tree T .

A finitely generated subgroup H < Fn is free, and its boundary ∂H of H isnaturally a closed subset of the boundary ∂Fn of Fn. If H stabilizes a point in Tthen the set ∂H × ∂H −∆ of pairs of distinct points in ∂H viewed as a subset of∂Fn × ∂Fn −∆ is contained in L2(T ).

We say that a leaf ℓ ∈ L2(T ) is carried by a finitely generated subgroup H of Fn

if it is a point in ∂H × ∂H −∆. A closed Fn-invariant subset C of ∂Fn × ∂Fn −∆intersects a free factor if there is a proper free factor H of Fn so that C ∩ ∂H ×∂H −∆ 6= ∅. We say that C is contained in H if C = Fn(C ∩ ∂H × ∂H −∆). Wealso say that a measured lamination µ is supported in a subgroup H of Fn if eachleaf of the support of µ is carried by a conjugate of H.

The following lemma is an immediate consequence of Theorem 4.5 of [R11]. Forits formulation, from now on we always denote by T the metric completion of anFn-tree T .

Lemma 6.1. Let T ∈ ∂cv(Fn) be indecomposable. If the zero lamination L2(T ) ofT intersects a free factor H then there is a point stabilizer in T which intersects a

free factor.

Proof. The intersection of L2(T ) with a free factor H is contained in the zerolamination of the minimal H-invariant subtree TH of T . By Theorem 4.5 of [R11],the action on T of any proper free factor H of Fn is discrete and hence TH issimplicial. Thus the zero lamination of TH is contained in a point stabilizer ofTH .

To obtain more information on supports of measured laminations µ with 〈T, µ〉 =0 we use the following consequence of Theorem 49 of [Ma95].


Lemma 6.2. Let H < Fn be a finitely generated subgroup of infinite index which

does not intersect a free factor. Then ∂H × ∂H −∆ does not support a measured

lamination for Fn.

Proof. Assume to the contrary that ∂H × ∂H −∆ supports an ergodic measuredlamination ν for Fn. Let ℓ ∈ ∂H × ∂H −∆ be a density point for ν.

Choose a free basis A for Fn and represent ℓ by a biinfinite word (wi) in thatbasis. Since H < Fn is finitely generated, H is quasiconvex and hence we mayassume that there is a sequence ni → ∞ such that each of the prefixes (wni

) of theword (wi) represent an element of H. By ergodicity of ν and the fact that ℓ is atypical point for ν, there is some i < j such that wnj

= wniwni

(the double of thesame word w = wni

).

We now follow the reasoning in the proof of Theorem 17 of [Ma95] (p.47).Namely, since H does not intersect a free factor, Lemma 48 and Theorem 49 of[Ma95] (which is attributed to Bestvina there) show that some Whitehead graphof w has no cut vertex and is connected. But every cyclic word v containing w2

as a subword has a Whitehead graph which has the Whitehead graph above asa subgraph, i.e. it has a connected Whitehead graph without cut vertices. ThusTheorem 49 of [Ma95] implies that a cyclically reduced word v containing w2 as asubword can not be primitive.

However, ν ∈ ML and hence ν is a weak∗-limit of measures µi which are dual toprimitive conjugacy classes for Fn [KL07]. This means in particular that the densitypoint ℓ for ν can be approximated by pairs of fixed points of primitive elements. Asa consequence, there is such a primitive element represented by a word in A whichcontains w2 as a subword. This is a contradiction which implies the lemma.

As a consequence of Lemma 6.1 and Lemma 6.2 we obtain

Corollary 6.3. An indecomposable tree [T ] is contained in FT if and only if no

point stabilizer for the action of Fn on T intersects a free factor.

Denote as before by T the metric completion of an Fn-tree T ∈ ∂cv(Fn) with

dense orbits. The union T = T ∪ ∂T of T with the Gromov boundary ∂T of Tcan be equipped with an observers’ topology. With respect to this topology, T is acompact Fn-space, and the inclusion T → T is continuous [CHL07]. Isometries of

T induce homeomorphisms of T (see p.903 of [CHL09]).

There is an explicit description of T as follows. Namely, let again L2(T ) be thezero lamination of T . There is an Fn-equivariant continuous map

Q : ∂Fn → T

[LL03] such that L2(T ) = (ξ, ζ) | Q(ξ) = Q(ζ) (Proposition 2.3 of [CHL07]).This map determines an equivariant homeomorphism

∂Fn/L2(T ) → T


(Corollary 2.6 of [CHL07]), i.e. the tree T is the quotient of ∂Fn by the equivalencerelation obtained by identifying all points ξ, ξ′ ∈ ∂Fn with Q(ξ) = Q(ξ′), and apair of points (ξ, ξ′) is identified if and only if it is contained in L2(T ).

Define a leaf ℓ ∈ L2(T ) to be regular if ℓ is not carried by the stabilizer of apoint in T and if moreover there exists a sequence ℓn ⊂ L2(T ) of leaves convergingto ℓ such that the xn = Q2(ℓn) are distinct. The set of regular leaves of L2(T ) isthe regular sublamination Lr(T ). It is Fn-invariant but in general not closed.

Recall from the introduction the definition of the set FT ⊂ ∂CV(Fn) of arationaltrees. A representative of a tree in the set FT is indecomposable.

We need a slight extension of a result of [CHR11]. In its formulation, we assumethat the action of Fn on an R-tree T is free, but we allow fixed points in the metriccompletion T . Thus there may be elliptic elements for the action of Fn on T . Recallthat a minimal subset of an Fn-space is an invariant set with each orbit dense.

Lemma 6.4. Let [T ] ∈ FT . If the Fn-action on T does not have fixed points then

Lr(T ) is minimal.

Proof. Since T is indecomposable, Proposition 5.14 of [CHR11] shows that T iseither of surface type or of Levitt type.

Consider first the case that T is of Levitt type. Since we defined Lr(T ) to consistof leaves not carried by a point stabilizer, Lemma 3.6 of [CHR11] (whose proof isvalid in the situation at hand) yields that if L0 ⊂ Lr(T ) is a proper sublaminationthen every leaf of L0 is carried by a proper free factor of Fn.

Let H be such a proper free factor which carries a leaf of L0. Since [T ] isindecomposable, by the main result of [R11] the minimal H-invariant subtree THof H is simplicial. Then TH is a very small simplicial H-tree with non-empty zerolamination. As a consequence, there is a nontrivial point stabilizer H0 for the actionof H on TH . Since H < Fn is a free factor, by [KL09] and Theorem 49 of [Ma95],the measure on ∂Fn × ∂Fn − ∆ which consists of Dirac masses at conjugates offixed points of a nontrivial element g ∈ H0 is a measured lamination µ supportedin H0. By [KL09], this measured lamination satisfies 〈T, µ〉 = 0. This violates thedefinition of a arational tree and hence indeed Lr(T ) is minimal.

In the case that T is of surface type the same reasoning applies. This time weuse Lemma 4.10 of [CHR11] whose proof is also valid in the situation at hand andconclude as before.

Let again [T ] ∈ FT and let µ be an ergodic measured lamination with supportSupp(µ) ⊂ L2(T ). By [KL09], this is equivalent to stating that 〈T, µ〉 = 0. For anypoint stabilizer H of T , a measured lamination supported in H is supported in thezero lamination of T . Thus by Lemma 6.2 and the definition of an arational tree,Supp(µ) is not contained in the stabilizer of a point in T . Then by ergodicity, µgives full measure to Lr(T ).

Define an equivalence relation ∼µ on ∂Fn as the smallest equivalence relationon ∂Fn which contains with ξ all points ξ′ so that (ξ, ξ′) ∈ Supp(µ). Let ∼ be


the closure of ∼µ. Note that this adds diagonal leaves to the support of µ. Byinvariance of Supp(µ) under the action of Fn, the quotient ∂Fn/ ∼ is a compactFn-space. Note also that the lamination L2(T ) defines an equivalence relationthanks to [CHL07].

Since by Corollary 2.6 of [CHL07] we have T = ∂Fn/L2(T ) and since Supp(µ) ⊂

L2(T ), there is a natural Fn-equivariant continuous surjection G : ∂Fn/ ∼→ T .

The next lemma shows that if [T ] ∈ FT then the tree T is uniquely determined bySupp(µ). It is an immediate consequence of the main result of [R12] (of which Ilearned after the first version of this paper was posted). We include a proof sinceit is not very difficult and illustrates some features of the observers’ topology.

Lemma 6.5. If [T ] ∈ FT then G is a homeomorphism.

Proof. If the action of Fn on T is free then the lemma is an immediate consequenceof the main result of [CHR11]. Namely, in this case the lamination Lr(T ) is minimal,and L2(T ) is the union of Lr(T ) and finitely many Fn-orbits of diagonal leaves. Nowdiagonals of Lr(T ) are always contained in the zero lamination of T (see [CHR11]and the above discussion). The closure of the Fn-orbit of a diagonal leaf ℓ iscontained in the union of ℓ with Lr(T ). An isolated leaf which is not defined bya pair of fixed points of some element of Fn can not be contained in the supportof a measured lamination. Thus the support of any measured lamination µ with〈T, µ〉 = 0 equals Lr(T ), and this is the complement of finitely many Fn-orbits ofdiagonal leaves in L2(T ). The lemma follows in this case.

Now assume that there is a nontrivial point stabilizer in T , i.e. that there is apoint x ∈ T whose stabilizer H is nontrivial. Recall that H is finitely generated[L94] and hence the boundary ∂H of H embeds into the boundary of Fn. If ξ ∈ ∂H

then Q(ξ) = x where Q : ∂Fn → T is the map introduced above (see [CH10] for adetailed discussion of this fact which is due to Coulbois, Hilion and Lustig).

We claim that x is contained in the closure of Q2(Supp(µ)) with respect to theobservers’ topology. Namely, let Ω = Q2(L2(T )) ⊂ T be the limit set of T andlet ξ ∈ Q2(Supp(µ)). Here the notation Q2 refers to applying the map Q to bothpoints which define ξ; the result is a single point in T . If ξ = x then we are done.Otherwise there is a segment ℓ in T connecting x to ξ. This segment defines adirection in T at x.

Let (hi) ⊂ H be a sequence of pairwise distinct elements. Since Hx = x,by invariance of Supp(µ) under the action of Fn and equivariance of the map Q,the images of the segment ℓ under the elements hi ∈ H defines a sequence ofsegments connecting x to points in Q2(Supp(µ)). Since edge stabilizers of T aretrivial [GL95], by passing to a subsequence we may assume that the correspondingsequence of directions at x are pairwise distinct. By the definition of the observers’topology, the points hi(ξ) converge as i→ ∞ in T to x (compare the discussion onp.903 of [CHL07]). Thus the point x is contained in the closure of the image of theFn-orbit of ξ for the observers’ topology and hence x = Q(∂H) is contained in theclosure of Q2(Supp(µ)).


There are now two cases. In the first case, every point x ∈ T with nontrivialpoint stabilizer is contained in T−T and the Fn-action on T is free. Lemma 6.4 andthe above discussion then show that the closure of Q2(Supp(µ)) for the observers’topology contains the entire limit set Q2(L2(T )). As a consequence, the closures ofQ2(Supp(µ)) and Q2(L2(T )) with respect to the observers’ topology coincide (notethat the limit set is in general not closed). Now ∂Fn/ ∼ is a compact Fn-space and

hence ∂Fn/ ∼= T which implies the lemma.

If there exists a fixed point x ∈ T then by invariance and the fact that T isdense we conclude that the tree T is contained in the closure of Q2(Supp(µ)) for

the observers’ topology. On the other hand, T is the interior of T for the observers’topology and therefore the interior of ∂Fn/ ∼ equals T . However, T is uniquelydetermined by T and hence G is an equivariant homeomorphism as claimed.

Remark: 1) Lemma 6.2 and Lemma 6.5 illustrate the fact that the observers’topology on the union of the metric completion of T with the Gromov boundary ofT is strictly weaker than the topology induced by the metric. We refer to [CHL07]for a detailed discussion.

2) The main result of [R12] gives a much more explicit description of an arationaltree. Namely, it states that an arational tree T either is free or dual to a measuredlamination of an oriented surface with a single boundary component.

Let ≈ be the smallest equivalence relation on ∂CV(Fn) whose classes containwith every [T ] ∈ ∂CV(Fn) all projections of [T ]. Lemma 6.5 is used to show

Corollary 6.6. Let [T ] ∈ FT and let µ be an ergodic measured lamination with

〈T, µ〉 = 0. Then [S] ∈ ∂CV(Fn) is equivalent to [T ] if and only if [S] ∈ FT and

〈S, µ〉 = 0.

Proof. Let [T ] ∈ FT and let µ ∈ ML be an ergodic measured lamination such that

〈T, µ〉 = 0. By Lemma 6.5, the tree T with the observers’ topology is determinedby Supp(µ). This means the following. If [S] is any tree with 〈S, µ〉 = 0 then there

is an equivariant continuous surjection T → S.

Since the interior of T , S, respectively, is just the tree T, S, Proposition 1.10 of[CHL07] shows that there is an alignment preserving map T → S. In other words,S is a projection of T and hence S is equivalent to T . Using once more the resultsof [R10] we conclude that [S] ∈ FT .

Recall that each conjugacy class of a primitive element g ∈ Fn determines ameasured lamination which is the set of all Dirac masses on the pairs of fixedpoints of the elements in the class. The measured lamination is called dual tothe conjugacy class. Recall also that the intersection form 〈, 〉 on cv(Fn) ×ML iscontinuous.

We have


Lemma 6.7. Let [Ti] ⊂ CV(Fn) be a sequence converging to some [T ] ∈ ∂CV(Fn).For each i let Ti ∈ cv0(Fn) be a representative of [Ti] and let αi be a primitive

short conjugacy class on Ti with dual measured lamination µi. If [T ] is dense then

up to passing to a subsequence, there is a sequence bi ⊂ (0, 1] such that the mea-

sured laminations biµi converge weakly to a measured lamination µ with 〈T, µ〉 = 0.Moreover, either µ is dual to a primitive conjugacy class or bi → 0.

Proof. Let T be a representative of [T ] and let ai ∈ (0,∞) be such that aiTi → Twhere Ti ∈ cv0(Fn). Since the Fn-orbits on T are dense, we have ai → 0 (i→ ∞).

Fix some tree S ∈ cv0(Fn). Then the set

Σ = ζ ∈ ML | 〈S, ζ〉 = 1

defines a section of the projection ML → PML. In particular, the space Σ iscompact.

Let µi be the lamination dual to a primitive short conjugacy class αi on Ti.There is a number ǫ > 0 so that 〈S, ζ〉 ≥ ǫ whenever ζ is dual to any primitiveconjugacy class. Thus if bi > 0 is such that biµi ∈ Σ then the sequence (bi) isbounded. Since Σ is compact, by passing to a subsequence we may assume thatbiµi → µ for some measured lamination µ ∈ Σ.

Now 〈aiTi, µi〉 ≤ kai where k ≥ 2 is as in Section 2 and hence since ai → 0(i→ ∞) and since the sequence (bi) is bounded, we have

〈aiTi, biµi〉 → 0 (i→ ∞).

The first part of the lemma now follows from continuity of the intersection form.Moreover, either bi → 0 or the length on S/Fn of the conjugacy classes αi is uni-formly bounded. However, there are only finitely many conjugacy classes of prim-itive elements which can be represented by a loop on S/Fn of uniformly boundedlength. Thus either bi → 0, or the sequence (αi) contains only finitely many el-ements and hence there is some primitive conjugacy class α so that αi = α forinfinitely many i. Then clearly µ is a multiple of the dual of α.

The next observation is a version of Proposition 6.4 of [K99].

Lemma 6.8. Let ([Ti]) ⊂ CV(Fn) be a sequence converging in CV(Fn)∪ ∂CV(Fn)to a point [T ]. If [T ] 6∈ FT then Υ(Ti) does not converge to a point in ∂FF .

Proof. We follow the reasoning in the proof of Proposition 6.4 of [K99]. Let ([Ti])be a sequence as in the lemma which converges to a point [T ] ∈ ∂CV(Fn)−FT .

By Proposition 4.1, Proposition 5.4, Corollary 5.9 and Lemma 6.6, we haveϕ∆([T ]) = Θ. More precisely, there is a proper free factor H of Fn so that forany Skora path (xt) ⊂ cv(Fn) converging to a representative T of [T ] and all largeenough t, the point xt is contained in a uniformly bounded neighborhood of H.

We argue by contradiction and we assume that the sequence Υ(Ti) converges toa point in the Gromov boundary of FF .


For fixed i and for j > i let rijt be an unnormalized Skora path connecting apoint of covolume one in a simplex ∆i defined by a basis containing the free factorΥ(Ti) to a tree Tj ∈ cv(Fn) which can be obtained from Tj by rescaling. Then the

trees Tj are contained in the compact set Σi ⊂ cv(Fn) of all trees which can bereached from a point in the simplex ∆i by a one-Lipschitz optimal map.

The initial points of the paths rijt are contained in a compact subset of cv(Fn) and

hence up to passing to a subsequence, we may assume that the paths rijt convergeas j → ∞ locally uniformly in Σi to a Skora path t → rt issueing from a point r0in the simplex ∆i. By the reasoning in the proof of Lemma 3.1, since [Ti] → [T ] wemay assume that the path (rt) is defined by a train track map r0 → T where T isa representative of [T ] and hence it connects r0 to T .

By Lemma 3.5 and Corollary 5.6, for large enough t the free factor Υ(rt) iscontained in a uniformly bounded neighborhood of the free factor H. Fix such apoint t0.

By construction, for large enough j there is a point riju on the Skora path (rijt )contained in a compact neigborhood of rt0 which is mapped by Υ to a set of

uniformly bounded diameter. This means that for large enough j the path Υ(rijt )passes through a uniformly bounded neighborhood of the free factor H.

Now the paths Υ(rijt ) are uniform unparametrized quasi-geodesics in FF . Asa consequence, if (|)H is the Gromov product based at H ∈ FF then we have(Υ(Ti) | Υ(Tj))H ≤ B for infinitely many i, j where B > 0 is a universal constant.This is a contradiction to the assumption that Υ(Ti) converges to a point in theGromov boundary of FF .

Next we have

Lemma 6.9. If [Ti] ⊂ CV(Fn) is any sequence which converges to [T ] ∈ FT then

Υ(Ti) converges to a point in ∂FF .

Proof. We show first that if [Ti] → [T ] ∈ FT then the sequence Υ(Ti) is unbounded.For this we use a variant of an argument of Luo as explained in [MM99].

Namely, let Ti ∈ cv0(Fn) be a representative of [Ti]. We argue by contradictionand we assume that after passing to a subsequence, the sequence Υ(Ti) remains ina bounded set in FF .

Since by Proposition 2.2 the map Υ is a quasi-isometry for the no-gap metric oncv0(Fn), after passing to another subsequence we may assume that for all i ≥ 1 thedistance between Ti and T0 in (cv0(Fn), dng) equals m for some m ≥ 0 which doesnot depend on i.

By the definition of the no-gap metric, this implies that for all i ≥ 1 there is asequence (Tj,i)0≤j≤m ⊂ cv0(Fn) with T0,i = T0 and Tm,i = Ti so that for all j < mthe trees Tj,i and T(j+1),i are tied. In particular, for each j < m there is a primitiveconjugacy class αj,i which can be represented by a curve of length at most k onboth Tj,i and T(j+1),i where k ≥ 2 is as in Section 2.


Let µj,i be the measured lamination which is dual to αj,i. By assumption, we

have [Tm,i] → [T ] (i → ∞) in CV(Fn). Since T is dense, Lemma 6.7 implies thatup to passing to a subsequence, there is a bounded sequence (bi) such that themeasured laminations biµm−1,i converge as i→ ∞ to a measured lamination νm−1

supported in the zero lamination of T . Since [T ] ∈ FT , the support of νm−1 doesnot intersect a free factor and hence bi → 0 by Lemma 6.7.

For each i let Ti ∈ cv0(Fn) be a representative of [Ti]. By passing to anothersubsequence, we may assume that the projective trees [T(m−1),i] converge as i →∞ to a projective tree [Um−1]. We claim that [Um−1] ∈ FT . Namely, choosea representative Um−1 of [Um−1]. Since biµm−1,i → νm−1 for a sequence bi →0 and since 〈Tm−1,i, µm−1,i〉 ≤ k for all i, we conclude from continuity of theintersection form that 〈Um−1, νm−1〉 = 0. In particular, νm−1 is supported in thezero lamination of Um−1. Lemma 6.6 now shows that [Um−1] ∈ FT . Moreover,there is a subsequence of the sequence [µm−2,i] which converges as i → ∞ to ameasured lamination supported in the zero lamination of [Um−1].

Repeat this argument with the sequence [T(m−2),i] and the tree [Um−1]. Afterm steps we conclude that [T0] ∈ FT which is impossible.

We are left with showing that the unbounded sequence Υ(Ti) converges to apoint in ∂FF . To this end assume to the contrary that the sequence Υ(Ti) doesnot converge to a point in ∂FF . Then we can find subsequences (xj), (yj) of thesequence (Ti) so that xj = Tu(j), yj = Tv(j) with u(j) → ∞, v(j) → ∞ as j → ∞

and such that for all j a Skora path rj connecting a point in the simplex of xj to yjpasses in the no gap metric through a uniformly bounded neighborhood of a basetree T0. Let pj be a point on this path which has this property.

Now if the points on the paths rj are normalized in such a way that the Lipschitzconstant of an optimal map from the fixed tree T0 to the points on rj equalsone and if αj is a primitive short loop on pj then using again continuity of theintersection form and compactness, up to passing to a subsequence the projectivemeasured laminations dual to αj converge to a projective measured lamination µwith 〈T, µ〉 = 0. By the first part of this proof, this implies that the sequence Υ(pj)is unbounded, a contradiction.

As an immediate consequence of Lemma 6.9 we obtain the analog of Proposition6.2 of [K99]. In the statement of the corollary, the set FT is equipped with thetopology as a subspace of ∂CV(Fn).

Corollary 6.10. The identity CV(Fn) → (CV(Fn), dng) extends to an Out(Fn)-equivariant continuous surjective map Y : FT → ∂FF .

Proof. By Lemma 6.8, Lemma 6.9 and compactness of CV(Fn), there is a surjectivemap which associates to a tree [T ] ∈ FT a point in ∂FF .

To show that this map is continuous, note that if Ti → T ∈ FT then thereis a sequence (rit) of Skora paths starting at a point in a standard simplex ∆so that rit → [Ti] (t → ∞) and ri → r locally uniformly where r is a Skorapath connecting ∆ to [T ]. From this, hyperbolicity of FF and coarse Lipschitz


continuity of Υ, continuity follows, and Out(Fn)-equivariance is immediate fromthe construction.

We are left with describing the fibres of the map Y and the Gromov topology on∂FF . To this end note first that the set of all trees [S] ∈ FT which are equivalentto a given tree [T ] ∈ FT is a closed subset of FT which is homeomorphic to thesimplex of projective length measures on [T ]. As a consequence, the equivalencerelation ∼ on FT defined above is closed, and the map Y factors through a con-tinuous bijection Λ : FT / ∼→ ∂FF where FT / ∼ is equipped with the quotienttopology.

Thus it now suffices to show the following. If [Ti] ⊂ FT is any sequence and ifΛ([Ti]) → Λ([T ]) then up to passing to a subsequence, we have [Ti] → [U ] where[U ] is equivalent to [T ].

Assume to the contrary that this is not the case. By compactness of CV(Fn)there is then a sequence [Ti] ⊂ FT so that Y [Ti] → Y [T ] and such that [Ti] →[S] ∈ ∂CV(Fn) where either [S] 6∈ FT or Y [S] 6= Y [T ].

Now if [S] ∈ FT then by Corollary 6.10, Y [S] = limi→∞ Y ([Ti]). Since weassumed that Y ([S]) 6= Y ([T ]) this is impossible. Thus [S] 6∈ FT . But this violatesLemma 6.8 and completes the proof of the theorem from the introduction.

We note an easy consequence which will be useful in other context. For itsformulation, following [H09] we call a pair (µ, ν) ∈ ML×ML positive if for every

tree T ∈ cv(Fn) we have 〈T, µ〉+ 〈T, ν〉 > 0.

Corollary 6.11. Let µ, ν ∈ ML be supported in the zero lamination of trees

[T ], [S] ∈ FT which determine different points in ∂FF . Then (µ, ν) is a posi-

tive pair.

In [H09] we defined an Fn-invariant set UT of projective trees in ∂CV(Fn) asfollows. If [T ] ∈ UT and if µ ∈ ML is such that 〈T, µ〉 = 0 then the projectiveclass of µ is unique, and [T ] is the unique projective tree with 〈T, µ〉 = 0. It followsimmediately from this work that UT ⊂ FT (see also [R12]).

Define the ǫ-thick part cv0(Fn)ǫ of cv0(Fn) to consist of simplicial trees withquotient of volume one which do not admit any essential loop of length smaller thanǫ. In analogy to properties of the curve graph and Teichmuller space, we conjecturethat whenever (rt) is a normalized Skora path in cv0(Fn) with the property thatrti ∈ cv0(Fn)ǫ for a sequence ti → ∞ and some fixed number ǫ > 0 then Υ(rt)converges as t→ ∞ to a point in ∂FF defined by a tree [T ] ∈ UT .

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Introduction - uni-bonn.de2 URSULA HAMENSTADT¨ Let cv(Fn) be the unprojectivized Outer space of minimal free simplicial Fn- trees, with its boundary ∂cv(Fn) of all minimal very