Commensurations of subgroups of Out(FN) - Horbez, C; Wade, R (American Mathematical Society, 2020)
A theorem of Farb and Handel [FH07] asserts that for N ≥ 4, the natural inclusion from Out(FN ) 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(FN ) ensuring that its abstract commensurator Comm(Γ) is isomorphic to its relative commensurator in Out(FN ). In particular, we prove that the abstract commensurator of the Torelli subgroup IAN for all N ≥ 3, or more generally any term of the Andreadakis–Johnson filtration if N ≥ 4, is equal to Out(FN ). Likewise, if Γ the kernel of the natural map from Out(FN ) to the outer automorphism group of a free Burnside group of rank N ≥ 3, then the natural map Out(FN ) → Comm(Γ) is an isomorphism.