A short proof of Handel and Mosher's alternative for subgroups of Out( F N )
aa r X i v : . [ m a t h . G R ] A p r A short proof of Handel and Mosher’s alternative forsubgroups of Out ( F N ) Camille HorbezOctober 11, 2018
Abstract
We give a short proof of a theorem of Handel and Mosher [18] stating that anyfinitely generated subgroup of Out ( F N ) either contains a fully irreducible automor-phism, or virtually fixes the conjugacy class of a proper free factor of F N , and weextend their result to non finitely generated subgroups of Out ( F N ) . Introduction
Let N ≥ , and let F N denote a finitely generated free group of rank N . A free factor of F N is a subgroup A of F N such that F N splits as a free product of the form F N = A ∗ B , forsome subgroup B ⊆ F N . An automorphism Φ ∈ Out ( F N ) is fully irreducible if no power of Φ preserves the conjugacy class of any proper free factor of F N . The goal of this paper is togive a short proof of the following classification theorem for subgroups of Out ( F N ) , whichwas shown by Handel and Mosher in the case of finitely generated subgroups of F N in [18]. Theorem 0.1.
Every (possibly non finitely generated) subgroup of Out ( F N ) either • contains two fully irreducible elements that generate a rank two free subgroup, or • is virtually cyclic, generated by a fully irreducible automorphism, or • virtually fixes the conjugacy class of a proper free factor of F N . Our proof of Theorem 0.1 involves studying the action of subgroups of Out ( F N ) on thefree factor complex, whose hyperbolicity was proved by Bestvina and Feighn in [4] (see also[29] for an alternative proof) and whose Gromov boundary was described by Bestvina andReynolds [6] and Hamenstädt [17]. We also use elementary tools that originally arose inthe study of random walks on groups, by studying stationary measures on the boundariesof outer space and of the free factor complex.Theorem 0.1 has already found various applications, for example to the study of mor-phisms from lattices to Out ( F N ) [7] or to spectral rigidity questions [9].1andel and Mosher have generalized Theorem 0.1 in a recent series of papers [19, 20,21, 22, 23] to give a complete classification of finitely generated subgroups of Out ( F N ) ,analogous to Ivanov’s classification of subgroups of the mapping class group of a finite typeoriented surface [25]. Acknowledgments
I warmly thank my advisor Vincent Guirardel for his numerous and helpful advice that ledto significant improvements in the exposition of the proof.
A geodesic metric space ( X, d ) is Gromov hyperbolic if there exists δ > such that forall x, y, z ∈ X , and all geodesic segments [ x, y ] , [ y, z ] and [ x, z ] , we have N δ ([ x, z ]) ⊆ N δ ([ x, y ]) ∪ N δ ([ y, z ]) (where given a subset Y ⊆ X and r ∈ R + , we denote by N r ( Y ) the r -neighborhood of Y in X ). The Gromov boundary ∂X of X is the space of equivalenceclasses of quasi-geodesic rays in X , two rays being equivalent if their images lie at boundedHausdorff distance. Isometry groups of Gromov hyperbolic spaces.
Let X be a hyperbolic geodesicmetric space. An isometry φ of X is loxodromic if for all x ∈ X , we have lim n → + ∞ n d ( x, φ n x ) > . Given a group G acting by isometries on X , we denote by ∂ X G the limit set of G in ∂X ,which is defined as the intersection of ∂X with the closure of the orbit of any point in X under the G -action. The following theorem, due to Gromov, gives a classification ofisometry groups of (possibly nonproper) Gromov hyperbolic spaces. The interested readerwill find a sketch of proof in [8, Proposition 3.1]. Theorem 1.1. (Gromov [14, Section 8.2]) Let X be a hyperbolic geodesic metric space,and let G be a group acting by isometries on X . Then G is either • bounded , i.e. all G -orbits in X are bounded; in this case ∂ X G = ∅ , or • horocyclic , i.e. G is not bounded and contains no loxodromic element; in this case ∂ X G is reduced to one point, or • lineal , i.e. G contains a loxodromic element, and any two loxodromic elements havethe same fixed points in ∂X ; in this case ∂ X G consists of these two points, or • focal , i.e. G is not lineal, contains a loxodromic element, and any two loxodromicelements have a common fixed point in ∂X ; in this case ∂ X G is uncountable and G has a fixed point in ∂ X G , or of general type , i.e. G contains two loxodromic elements with no common endpoints;in this case ∂ X G is uncountable and G has no finite orbit in ∂X . In addition, thegroup G contains two loxodromic isometries that generate a rank two free subgroup. In particular, we have the following result.
Theorem 1.2. (Gromov [14, Section 8.2]) Let X be a hyperbolic geodesic metric space,and let G be a group acting by isometries on X . If ∂ X G = ∅ , and G has no finite orbit in ∂X , then G contains a rank two free subgroup generated by two loxodromic isometries. Let N ≥ . Outer space CV N is defined to be the space of simplicial free, minimal, isometricactions of F N on simplicial metric trees, up to F N -equivariant homotheties [12] (an actionof F N on a tree is minimal if there is no proper invariant subtree). We denote by cv N the unprojectivized outer space , in which trees are considered up to equivariant isometries,instead of homotheties. The group Out ( F N ) acts on CV N and on cv N on the right byprecomposing the actions (one can also consider the Out ( F N ) -action on the left by setting Φ( T, ρ ) = (
T, ρ ◦ φ − ) for all Φ ∈ Out ( F N ) , where ρ : F N → Isom ( T ) denotes the action,and φ ∈ Aut ( F N ) is any lift of Φ to Aut ( F N ) ).An R -tree is a metric space ( T, d T ) in which any two points x and y are joined by aunique arc, which is isometric to a segment of length d T ( x, y ) . Let T be an F N -tree , i.e. an R -tree equipped with an isometric action of F N . For g ∈ F N , the translation length of g in T is defined to be || g || T := inf x ∈ T d T ( x, gx ) . Culler and Morgan have shown in [11, Theorem 3.7] that the map i : cv N → R F N T ( || g || T ) g ∈ F N is an embedding, whose image projects to a subspace of PR F N with compact closure CV N [11, Theorem 4.5]. Bestvina and Feighn [2], extending results by Cohen and Lustig [10],have characterized the points of this compactification as being the minimal F N -trees withtrivial or maximally cyclic arc stabilizers and trivial tripod stabilizers. The free factor complex
F F N , introduced by Hatcher and Vogtmann in [24], is defined when N ≥ as the simplicial complex whose vertices are the conjugacy classes of nontrivial properfree factors of F N , and higher dimensional simplices correspond to chains of inclusions offree factors. (When N = 2 , one has to modify this definition by adding an edge betweenany two complementary free factors to ensure that F F remains connected, and F F isisomorphic to the Farey graph). Gromov hyperbolicity of F F N was proved by Bestvina andFeighn [4] (see also [29] for an alternative proof). There is a natural, coarsely well-definedmap ψ : CV N → F F N , that maps any tree T ∈ CV N to one of the conjugacy classes ofthe cyclic free factors of F N generated by an element of F N whose axis in T projects to an3mbedded simple loop in the quotient graph T /F N . The Gromov boundary of F F N wasdetermined independently by Bestvina and Reynolds [6] and by Hamenstädt [17]. A tree T ∈ ∂CV N is arational if no proper free factor of F N acts with dense orbits on its minimalsubtree in T (in particular, no proper free factor of F N is elliptic in T ). We denote by AT the subspace of ∂CV N consisting of arational trees. We define an equivalence relation ∼ on AT by setting T ∼ T ′ whenever T and T ′ have the same underlying topological tree. Theorem 1.3. (Bestvina-Reynolds [6], Hamenstädt [17]) There is a unique homeomor-phism ∂ψ : AT / ∼→ ∂ F F N , so that for all T ∈ AT and all sequences ( T n ) n ∈ N ∈ CV N N that converge to T , the sequence ( ψ ( T n )) n ∈ N converges to ∂ψ ( T ) . Recall from the introduction that an automorphism Φ ∈ Out ( F N ) is fully irreducible if no nonzero power of Φ preserves the conjugacy class of any proper free factor of F N .Bestvina and Feighn have characterized elements of Out ( F N ) which act as loxodromicisometries of F F N . Theorem 1.4. (Bestvina-Feighn [4, Theorem 9.3]) An outer automorphism Φ ∈ Out ( F N ) acts loxodromically on F F N if and only if it is fully irreducible. Given T ∈ ∂CV N r AT , the set of conjugacy classes of minimal (with respect to inclu-sion) proper free factors of F N which act with dense orbits on their minimal subtree in T , but are not elliptic in T , is finite [32, Corollary 7.4 and Proposition 9.2], and dependsOut ( F N ) -equivariantly on T . We denote it by Dyn ( T ) . The following proposition essen-tially follows from Reynolds’ arguments in his proof of [32, Theorem 1.1], we provide asketch for completeness. Proposition 2.1.
For all T ∈ ∂CV N r AT , either Dyn ( T ) = ∅ , or there is a nontrivialpoint stabilizer in T which is contained in a proper free factor of F N . In the proof of Proposition 2.1, we will make use of the following well-known fact.
Lemma 2.2. (see [32, Corollary 11.2]) Let T be a simplicial F N -tree, all of whose edgestabilizers are (at most) cyclic. Then every edge stabilizer in T is contained in a proper freefactor of F N , and there is at most one conjugacy class of vertex stabilizers in T that is notcontained in any proper free factor of F N .Proof of Proposition 2.1. Let T ∈ ∂CV N r AT , and assume that Dyn ( T ) = ∅ . First assumethat T contains an edge with nontrivial stabilizer. Let S be the simplicial tree obtained bycollapsing all vertex trees to points in the decomposition of T as a graph of actions definedin [30]. Then edge stabilizers in T are also edge stabilizers in S , and the conclusion followsfrom Lemma 2.2.Otherwise, as in the proof of [32, Proposition 10.3], we get that if some point stabilizerin T is not contained in any proper free factor of F N , then T is geometric, has dense orbits,and all its minimal components are surfaces (the reader is referred to [3, 13] for backgroundon geometric F N -trees). Dual to the decomposition of T into its minimal components is abipartite simplicial F N -tree S called the skeleton of T , defined as follows [16, Section 1.3].4ertices of S are of two kinds: some correspond to minimal components Y of T , and theothers correspond to points x ∈ T belonging to the intersection of two distinct minimalcomponents. There is an edge from the vertex associated to x to the vertex associated to Y whenever x ∈ Y . In particular, point stabilizers in S are either point stabilizers in T ,or groups acting with dense orbits on their minimal subtree in T . In a minimal surfacecomponent, all point stabilizers are cyclic, so S is a simplicial F N -tree with (at most) cyclicedge stabilizers. If S is nontrivial, Lemma 2.2 implies that either a point stabilizer in T is contained in a proper free factor, or some subgroup of F N is contained in a proper freefactor F of F N and acts with dense orbits on its minimal subtree in T . In the latter case, bydecomposing the F -action on the F -minimal subtree of T as a graph of actions with trivialarc stabilizers [30], we get that Dyn ( T ) = ∅ , which has been excluded. If S is reduced to apoint, then T is minimal and dual to a surface with at least two boundary curves (otherwise T would be arational by [32, Theorem 1.1]). Any of these curves yields the desired pointstabilizer in T . ( F N ) A subgroup H ⊆ Out ( F N ) is nonelementary if it does not preserve any finite set of F F N ∪ ∂ F F N . In this section, we will prove Theorem 0.1 for nonelementary subgroups of Out ( F N ) . Theorem 3.1.
Every nonelementary subgroup of Out ( F N ) contains a rank two free sub-group, generated by two fully irreducible automorphisms. Stationary measures on ∂CV N . Our proof of Theorem 3.1 is based on techniques thatoriginally arose in the study of random walks on groups. All topological spaces will beequipped with their Borel σ -algebra. Let µ be a probability measure on Out ( F N ) . Aprobability measure ν on CV N is µ -stationary if µ ∗ ν = ν , i.e. for all ν -measurable subsets E ⊆ CV N , we have ν ( E ) = X Φ ∈ Out ( F N ) µ (Φ) ν (Φ − E ) . Our first goal will be to prove the following fact.
Proposition 3.2.
Let µ be a probability measure on Out ( F N ) , whose support generates anonelementary subgroup of Out ( F N ) . Then every µ -stationary probability measure on CV N is supported on AT . We will make use of the following classical lemma, whose proof is based on a maxi-mum principle argument (we provide a sketch for completeness). We denote by gr ( µ ) thesubgroup of Out ( F N ) generated by the support of the measure µ . Lemma 3.3. (Ballmann [1], Woess [33, Lemma 3.4], Kaimanovich-Masur [27, Lemma2.2.2]) Let µ be a probability measure on a countable group G , and let ν be a µ -stationaryprobability measure on a G -space X . Let D be a countable G -set, and let Θ : X → D be a measurable G -equivariant map. If E ⊆ X is a G -invariant measurable subset of X satisfying ν ( E ) > , then Θ( E ) contains a finite gr ( µ ) -orbit. roof. Let e ν be the probability measure on D defined by setting e ν ( Y ) := ν (Θ − ( Y )) forall subsets Y ⊆ D . It follows from µ -stationarity of ν and G -equivariance of Θ that e ν is µ -stationary. Let M ⊆ Θ( E ) denote the set consisting of all x ∈ Θ( E ) such that e ν ( x ) ismaximal (and in particular positive). Since e ν is a probability measure, the set M is finiteand nonempty. For all x ∈ M , we have e ν ( x ) = X g ∈ G µ ( g ) e ν ( g − x ) ≤ e ν ( x ) X g ∈ G µ ( g ) = e ν ( x ) , which implies that for all g ∈ G belonging to the support of µ , we have e ν ( g − x ) = e ν ( x ) .Therefore, the set M is invariant under the semigroup generated by the support of ˇ µ (where ˇ µ ( g ) := µ ( g − ) ). As M is finite, this implies that M is gr ( µ ) -invariant, so it contains afinite gr ( µ ) -orbit.We now define an Out ( F N ) -equivariant map Θ from CV N to the (countable) set D offinite collections of conjugacy classes of proper free factors of F N . Given a tree T ∈ CV N ,we define Loop ( T ) to be the finite collection of conjugacy classes of elements of F N whoseaxes in T project to an embedded simple loop in the quotient graph T /F N (these maybe viewed as cyclic free factors of F N ). Given T ∈ CV N , the set of conjugacy classes ofpoint stabilizers in T is finite [26]. Every point stabilizer is contained in a unique minimal(possibly non proper) free factor of F N , defined as the intersection of all free factors of F N containing it (the intersection of a family of free factors of F N is again a free factor). Welet Per ( T ) be the (possibly empty) finite set of conjugacy classes of proper free factors of F N that arise in this way, and we set Θ( T ) := ∅ if T ∈ AT Loop ( T ) if T ∈ CV N Dyn ( T ) ∪ Per ( T ) if T ∈ ∂CV N r AT . Proposition 2.1 implies that Θ( T ) = ∅ if and only if T ∈ AT . Lemma 3.4.
The set AT is measurable, and Θ is measurable. We postpone the proof of Lemma 3.4 to the next paragraph and first explain how todeduce Proposition 3.2.
Proof of Proposition 3.2.
Nonelementarity of gr ( µ ) implies that the only finite gr ( µ ) -orbitin D is the orbit of the empty set. Therefore, since Θ( T ) = ∅ as soon as T ∈ CV N r AT (Proposition 2.1), the set Θ( CV N r AT ) contains no finite gr ( µ ) -orbit. Proposition 3.2then follows from Proposition 3.3. Measurability of Θ . Given a finitely generated subgroup F of F N , we denote by P ( F ) the set of trees T ∈ CV N in which F is elliptic, by E ( F ) the set of trees T ∈ CV N in which F fixes an edge, and by D ( F ) the set of trees T ∈ CV N whose F -minimal subtree is anontrivial F -tree with dense orbits. Lemma 3.5.
For all finitely generated subgroups F ⊆ F N , the sets P ( F ) , E ( F ) and D ( F ) are measurable. roof. Let F be a finitely generated subgroup of F N . Let s : CV N → cv N be a continuoussection. We have P ( F ) = \ w ∈ F { T ∈ CV N ||| w || s ( T ) = 0 } , so P ( F ) is measurable. An element g ∈ F N fixes an arc in a tree T ∈ CV N if and onlyif it is elliptic, and there exist two hyperbolic isometries h and h ′ of T whose translationaxes both meet the fixed point set of g but are disjoint from each other. These conditionscan be expressed in terms of translation length functions: they amount to requiring that || gh || s ( T ) ≤ || h || s ( T ) and || gh ′ || s ( T ) ≤ || h ′ || s ( T ) and || hh ′ || s ( T ) > || h || s ( T ) + || h ′ || s ( T ) , see [11,1.5]. So E ( F ) is a measurable set, too.The F -minimal subtree of a tree T ∈ CV N has dense orbits if and only if for all n ∈ N , there exists a free basis { s , . . . , s k } of F so that for all i, j ∈ { , . . . , k } , we have || s i || s ( T ) ≤ n and || s i s j || s ( T ) ≤ n . This implies that the set Dense ( F ) consisting of thosetrees in CV N whose F -minimal subtree has dense orbits is measurable. Therefore D ( F ) = Dense ( F ) ∩ c P ( F ) is also measurable. Proof of Lemma 3.4.
Measurability of AT follows from Lemma 3.5. For all T ∈ CV N ,the set Dyn ( T ) consists of conjugacy classes of minimal free factors of F N that act withdense orbits on their minimal subtree in T but are not elliptic, so measurability of the map T Dyn ( T ) follows from measurability of D ( F ) for all finitely generated subgroups F of F N . Point stabilizers in T are either maximal among elliptic subgroups, or fix an arc in T .Therefore, since P ( F ) and E ( F ) are measurable for all finitely generated subgroups F of F N , the set of conjugacy classes of point stabilizers in a tree T ∈ CV N depends measurablyon T . Measurability of T Per ( T ) follows from this observation. As open simplices in CV N are also measurable, measurability of Θ follows. End of the proof of Theorem 3.1.Proposition 3.6.
Let H ⊆ Out ( F N ) be a nonelementary subgroup of Out ( F N ) . Then the H -orbit of any point x ∈ CV N has a limit point in AT .Proof. Let µ be a probability measure on Out ( F N ) whose support generates H . Since CV N is compact, the sequence of convolutions ( µ ∗ n ∗ δ x ) n ∈ N has a weak- ∗ limit point ν , whichis a µ -stationary measure on CV N . We have ν ( Hx ) = 1 , where Hx denotes the H -orbitof x in CV N , and Proposition 3.2 implies that ν ( AT ) = 1 . This shows that Hx ∩ AT isnonempty.As a consequence of Theorem 1.3 and Proposition 3.6, we get the following fact. Corollary 3.7.
Let H ⊆ Out ( F N ) be a nonelementary subgroup of Out ( F N ) . Then the H -orbit of any point in F F N has a limit point in ∂ F F N .Proof of Theorem 3.1. Let H be a nonelementary subgroup of Out ( F N ) . Corollary 3.7shows that the H -orbit of any point in F F N has a limit point in ∂ F F N . As H has nofinite orbit in ∂ F F N , Theorem 1.2 shows that H contains two loxodromic isometries whichgenerate a free group of rank two. Theorem 3.1 then follows from the fact that elements ofOut ( F N ) that act loxodromically on F F N are fully irreducible (Theorem 1.4).7 Proof of Theorem 0.1
Let H be a subgroup of Out ( F N ) . If H is nonelementary, then the claim follows fromTheorem 3.1. Otherwise, either H fixes a finite subset of conjugacy classes of proper freefactors (in which case a finite index subgroup of H fixes the conjugacy class of a properfree factor of F N ), or H virtually fixes a point in ∂ F F N . The set of trees in ∂CV N thatproject to this point is a finite-dimensional simplex in ∂CV N by [15, Corollary 5.4], and H fixes the finite subset of extremal points of this simplex. Up to passing to a finite indexsubgroup again, we can assume that H fixes an arational tree T ∈ ∂CV N . By Reynolds’characterization of arational trees [32, Theorem 1.1], either T is free, or else T is dual toan arational measured lamination on a surface S with one boundary component. In thefirst case, it follows from [28, Theorem 1.1] that H is virtually cyclic, virtually generatedby an automorphism Φ ∈ Out ( F N ) , and in this case Φ is fully irreducible, otherwise H would virtually fix the conjugacy class of a proper free factor of F N . In the second case,all automorphisms in H can be realized as diffeomorphisms of S [5, Theorem 4.1]. So H is a subgroup of the mapping class group of S , and the claim follows from the analogousclassical statement that stabilizers of arational measured foliations are virtually cyclic [31,Proposition 2.2]. References [1] W. Ballmann,
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