Hyperbolic one-relator groups
We introduce two families of two-generator one-relator groups called primitive extension groups and show that a one-relator group is hyperbolic if its primitive extension subgroups are hyperbolic. This reduces the problem of characterizing hyperbolic one-relator groups to characterizing hyperbolic p...
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| 格式: | Journal article |
| 語言: | English |
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Cambridge University Press
2024
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| _version_ | 1826317183693619200 |
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| author | Linton, M |
| author_facet | Linton, M |
| author_sort | Linton, M |
| collection | OXFORD |
| description | We introduce two families of two-generator one-relator groups called primitive extension groups and show that a one-relator group is hyperbolic if its primitive extension subgroups are hyperbolic. This reduces the problem of characterizing hyperbolic one-relator groups to characterizing hyperbolic primitive extension groups. These new groups, moreover, admit explicit decompositions as graphs of free groups with adjoined roots. In order to obtain this result, we characterize $\mathcal{2}$-free one-relator groups with exceptional intersection in terms of Christoffel words, show that hyperbolic one-relator groups have quasi-convex Magnus subgroup, and build upon the one-relator tower machinery developed in previous work of the author. |
| first_indexed | 2024-09-25T04:24:28Z |
| format | Journal article |
| id | oxford-uuid:3d907dc9-a8d7-4ad6-a83d-b1ac78acf714 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2025-02-19T04:36:01Z |
| publishDate | 2024 |
| publisher | Cambridge University Press |
| record_format | dspace |
| spelling | oxford-uuid:3d907dc9-a8d7-4ad6-a83d-b1ac78acf7142025-01-31T10:26:36ZHyperbolic one-relator groupsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:3d907dc9-a8d7-4ad6-a83d-b1ac78acf714EnglishSymplectic ElementsCambridge University Press2024Linton, MWe introduce two families of two-generator one-relator groups called primitive extension groups and show that a one-relator group is hyperbolic if its primitive extension subgroups are hyperbolic. This reduces the problem of characterizing hyperbolic one-relator groups to characterizing hyperbolic primitive extension groups. These new groups, moreover, admit explicit decompositions as graphs of free groups with adjoined roots. In order to obtain this result, we characterize $\mathcal{2}$-free one-relator groups with exceptional intersection in terms of Christoffel words, show that hyperbolic one-relator groups have quasi-convex Magnus subgroup, and build upon the one-relator tower machinery developed in previous work of the author. |
| spellingShingle | Linton, M Hyperbolic one-relator groups |
| title | Hyperbolic one-relator groups |
| title_full | Hyperbolic one-relator groups |
| title_fullStr | Hyperbolic one-relator groups |
| title_full_unstemmed | Hyperbolic one-relator groups |
| title_short | Hyperbolic one-relator groups |
| title_sort | hyperbolic one relator groups |
| work_keys_str_mv | AT lintonm hyperboliconerelatorgroups |