2019/01/22 by Camille Horbez, Horbez, Camille, Richard D. Wade +1
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.1901.07433
A theorem of Farb and Handel 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)\toComm(Γ) is an isomorphism.