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...
मुख्य लेखकों: | , |
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स्वरूप: | Journal article |
भाषा: | English |
प्रकाशित: |
American Mathematical Society
2020
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_version_ | 1826309310782636032 |
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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. |
first_indexed | 2024-03-07T07:33:46Z |
format | Journal article |
id | oxford-uuid:b3d4087b-9f82-483f-a9f5-dd28063715c9 |
institution | University of Oxford |
language | English |
last_indexed | 2024-03-07T07:33:46Z |
publishDate | 2020 |
publisher | American Mathematical Society |
record_format | dspace |
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 |