Realising all countable groups as quasi-isometry groups
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911421599580160 |
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| author | Heim, Paula MacManus, Joseph Mineh, Lawk |
| author_facet | Heim, Paula MacManus, Joseph Mineh, Lawk |
| contents | Given any countable group $G$, we construct uncountably many quasi-isometry classes of proper geodesic metric spaces with quasi-isometry group isomorphic to $G$. Moreover, if the group $G$ is a hyperbolic group, the spaces we construct are hyperbolic metric spaces.
We make use of a rigidity phenomenon for quasi-isometries exhibited by many symmetric spaces, called strong quasi-isometric rigidity. Our method involves the construction of new examples of strongly quasi-isometrically rigid spaces, arising as graphs of strongly quasi-isometrically rigid rank-one symmetric spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06261 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Realising all countable groups as quasi-isometry groups Heim, Paula MacManus, Joseph Mineh, Lawk Group Theory Metric Geometry 20F65 (Primary) 51F30 (Secondary) Given any countable group $G$, we construct uncountably many quasi-isometry classes of proper geodesic metric spaces with quasi-isometry group isomorphic to $G$. Moreover, if the group $G$ is a hyperbolic group, the spaces we construct are hyperbolic metric spaces. We make use of a rigidity phenomenon for quasi-isometries exhibited by many symmetric spaces, called strong quasi-isometric rigidity. Our method involves the construction of new examples of strongly quasi-isometrically rigid spaces, arising as graphs of strongly quasi-isometrically rigid rank-one symmetric spaces. |
| title | Realising all countable groups as quasi-isometry groups |
| topic | Group Theory Metric Geometry 20F65 (Primary) 51F30 (Secondary) |
| url | https://arxiv.org/abs/2601.06261 |