Commensurations of subgroups of Out(FN)

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 subgro...

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मुख्य लेखकों: Horbez, C, Wade, R
स्वरूप: Journal article
भाषा:English
प्रकाशित: American Mathematical Society 2020
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author Horbez, C
Wade, R
author_facet Horbez, C
Wade, R
author_sort Horbez, C
collection OXFORD
description 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.
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spelling oxford-uuid:b3d4087b-9f82-483f-a9f5-dd28063715c92023-02-09T08:48:28ZCommensurations of subgroups of Out(FN)Journal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:b3d4087b-9f82-483f-a9f5-dd28063715c9EnglishSymplectic Elements at OxfordAmerican Mathematical Society2020Horbez, CWade, RA 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.
spellingShingle Horbez, C
Wade, R
Commensurations of subgroups of Out(FN)
title Commensurations of subgroups of Out(FN)
title_full Commensurations of subgroups of Out(FN)
title_fullStr Commensurations of subgroups of Out(FN)
title_full_unstemmed Commensurations of subgroups of Out(FN)
title_short Commensurations of subgroups of Out(FN)
title_sort commensurations of subgroups of out fn
work_keys_str_mv AT horbezc commensurationsofsubgroupsofoutfn
AT wader commensurationsofsubgroupsofoutfn