Commensurations of subgroups of $\mathrm{Out}(F_N)$

TL;DR

This paper proves new results about the structure of commensurators of subgroups of Out(F_N), extending previous theorems to N=3 and identifying conditions under which the commensurator equals the ambient group.

Contribution

It provides a new proof of Farb and Handel's theorem and generalizes it to include N=3, also characterizes the commensurators of various important subgroups of Out(F_N).

Findings

The abstract commensurator of Out(F_N) is isomorphic to Out(F_N) for N≥4.

The result is extended to N=3, filling a previous gap.

The commensurator of the Torelli subgroup and certain filtrations equals Out(F_N).

Abstract

A theorem of Farb and Handel asserts that for $N\ge 4$, the natural inclusion from $\mathrm{Out}(F_N)$ 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 $\Gamma$ of $\mathrm{Out}(F_N)$ ensuring that its abstract commensurator $\mathrm{Comm}(\Gamma)$ is isomorphic to its relative commensurator in $\mathrm{Out}(F_N)$. In particular, we prove that the abstract commensurator of the Torelli subgroup $\mathrm{IA}_N$ for all $N\ge 3$, or more generally any term of the Andreadakis--Johnson filtration if $N\ge 4$, is equal to $\mathrm{Out}(F_N)$. Likewise, if $\Gamma$ the kernel of the natural map from $\mathrm{Out}(F_N)$ to the outer automorphism group of a free Burnside group of rank $N\geq 3$, then the natural map $\mathrm{Out}(F_N)\to\mathrm{Comm}(\Gamma)$ is an isomorphism.