A short proof of Handel and Mosher's alternative for subgroups of Out$(F_N)$

Abstract

We give a short proof of a theorem of Handel and Mosher [18] stating that any finitely generated subgroup of Out(F N ) either contains a fully irreducible automorphism, or virtually fixes the conjugacy class of a proper free factor of F N , and we extend their result to non finitely generated subgroups of Out(F N ).

The Poisson boundary of Out(FN)

“…A subgroup H ⊆ Out(F N ) is nonelementary if the H-orbits of all proper free factors of F N , of all projective arational trees, and of all conjugacy classes of elements of F N , are infinite. Arguing as in [27] (see also [25]), one can show that this is equivalent to H containing two independent atoroidal fully irreducible elements (we will not use this fact in the sequel). The following result is a consequence of Propositions 2.6, 2.8 and 2.15.…”

“…The following proposition was essentially proved in [27,Proposition 3.2], without the assumption that gr(µ) does not preserve any finite set of conjugacy classes of elements of F N . By the same reasoning as in the proof in [27], we will show this extra assumption implies that every µ-stationary measure is concentrated on the set of free actions. Measurability of AT was proved in [27,Lemma 3.4], and measurability of FI follows since freeness of the action is a measurable condition.…”

Distortion and Tits alternative in smooth mapping class groups

“…The following result is the key lemma for proving the existence of such an element h sending Per 0 (f ) ∪ U f disjoint from itself. It is based on a recent proof by Camille Horbez (See [8], Sec.3, [9]) of the Tits alternative for mapping class groups, outer automorphisms of free groups and other related groups.…”

“…The proof of the previous lemma is based on Horbez's recent proof of the Tits alternative for mapping class groups and related automorphisms groups (see [8] and [9]).…”

A train track directed random walk on Out(Fr)

“…For example, the Tits Alternative for Out(F r ), established in full generality in [BFH00,BFH05], was first proved in [BFH97] for subgroups of Out(F r ) containing a fully irreducible element. A result of Handel and Mosher [HM09], with a recent different proof by Horbez [Hor14b], shows that if H ≤ Out(F r ) is a finitely generated subgroup, then either H contains a fully irreducible element or H contains a subgroup H 1 of finite index in H such that H 1 preserves the conjugacy class of a proper free factor of F r . Also, fully irreducible elements are known to have particularly nice properties for the natural actions of Out(F r ) on various spaces.…”