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

Abstract:

A theorem of Farb and Handel [Publ. Math. Inst. Hautes Études Sci. 105 (2007), pp. 1–48] 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 $\Gamma$ is 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.