Author
Horbez, C
Wade, R
Journal title
Transactions of the American Mathematical Society
DOI
10.1090/tran/7991
Issue
4
Volume
373
Last updated
2024-02-27T13:18:07.03+00:00
Abstract
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.
Symplectic ID
983287
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Publication type
Journal Article
Publication date
23 Jan 2020
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