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...
| Main Authors: | , |
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| Formato: | Journal article |
| Idioma: | English |
| Publicado em: |
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 |