Commensurations of ${{\rm{Aut}}}(F_N)$ and its Torelli subgroup

TL;DR

This paper proves that for all N ≥ 3, the abstract commensurators of the automorphism group of a free group and its Torelli subgroup are isomorphic to the automorphism group itself, revealing a rigidity property.

Contribution

It establishes the isomorphism between the abstract commensurators and the automorphism group for both Aut(F_N) and IA_N, extending known rigidity results.

Findings

Abstract commensurators of Aut(F_N) are isomorphic to Aut(F_N) for N ≥ 3.

Abstract commensurators of IA_N are isomorphic to Aut(F_N) for N ≥ 3.

Shows rigidity of automorphism groups of free groups and their Torelli subgroups.

Abstract

For $N \geq 3$, the abstract commensurators of both ${{\rm{Aut}}}(F_N)$ and its Torelli subgroup ${{\rm{IA}}}_N$ are isomorphic to ${{\rm{Aut}}}(F_N)$ itself.