Commensurations of subgroups of 𝑂𝑢𝑡(𝐹_{𝑁})

Abstract

A theorem of Farb and Handel [FH07] asserts that for N ≥ 4, the natural inclusion from Out(F N ) into its abstract commensurator is an isomorphism. We give a new proof of their result, which enables us to generalize it to the case where N = 3. More generally, we give sufficient conditions on a subgroup Γ of Out(F N ) ensuring that its abstract commensurator Comm(Γ) is isomorphic to its relative commensurator in Out(F N ). In particular, we prove that the abstract commensurator of the Torelli subgroup IA N for all N ≥ 3, or more generally any term of the Andreadakis-Johnson filtration if N ≥ 4, is equal to Out(F N ). Likewise, if Γ the kernel of the natural map from Out(F N ) to the outer automorphism group of a free Burnside group of rank N ≥ 3, then the natural map Out(F N ) → Comm(Γ) is an isomorphism.

Commensurations of the outer automorphism group of a universal Coxeter group

“…Then the group of twists of S is isomorphic to Z by a result of Levitt (see [Lev,Proposition 3.1]). The next proposition is similar to a result in the case of the outer automorphism group of a free group (see [CL2] and [HW,Lemma 2.7]). Recall that an element w P W n is root-closed if there does not exist w 0 P W n and an integer n ě 2 such that w " w n 0 .…”

“…Proposition 2.1. [HW,Proposition 1.1] Let G be a group. Let X be a graph with no edge-loops, no multiple edges between pairs of vertices and such that G acts on X by graph automorphisms.…”

“…Lemma 5.2. [HW,Lemma 6.3] Let 1 Ñ N Ñ G Ñ Q Ñ 1 be a short exact sequence of groups. Then rk prod pGq ď rk prod pN q `rk prod pQq.…”

“…Lemma 5.3. [HW,Proposition 4.2,Lemma 4.4] Let X be a Gromov hyperbolic space, and let H be a group acting by isometries on X. Assume that H contains a normal subgroup K isomorphic to a direct product K " ś k i"1 K i .…”

“…Proof of Theorem 5.1. The proof is inspired by [HW,Theorem 6.1] due to Horbez and Wade and [HHW,Theorem 4.3] due to Hensel, Horbez and Wade. We first prove that if n ě 4, then rk prod pOutpW n qq ě n ´3 and that, if n ě 3, then rk prod pAutpW n qq ě n ´2.…”