Drilling hyperbolic groups
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Groves, Daniel Haïssinsky, Peter Manning, Jason F. Osajda, Damian Sisto, Alessandro Walsh, Genevieve S. |
| author_facet | Groves, Daniel Haïssinsky, Peter Manning, Jason F. Osajda, Damian Sisto, Alessandro Walsh, Genevieve S. |
| contents | Given a hyperbolic group $G$ and a maximal infinite cyclic subgroup $\langle g \rangle$, we define a {\it drilling of $G$ along $g$}, which is a relatively hyperbolic group pair $(\widehat{G}, P)$. This is inspired by the well-studied procedure of drilling a hyperbolic $3$--manifold along an embedded geodesic. We prove that, under suitable conditions, a hyperbolic group with $2$-sphere boundary admits a drilling where the resulting relatively hyperbolic group pair $(\widehat{G}, P)$ has relatively hyperbolic boundary $S^2$. This allows us to reduce the Cannon Conjecture (in the residually finite case) to a relative version, which is likely to be more tractable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14667 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Drilling hyperbolic groups Groves, Daniel Haïssinsky, Peter Manning, Jason F. Osajda, Damian Sisto, Alessandro Walsh, Genevieve S. Geometric Topology Group Theory 20F67, 20F65 Given a hyperbolic group $G$ and a maximal infinite cyclic subgroup $\langle g \rangle$, we define a {\it drilling of $G$ along $g$}, which is a relatively hyperbolic group pair $(\widehat{G}, P)$. This is inspired by the well-studied procedure of drilling a hyperbolic $3$--manifold along an embedded geodesic. We prove that, under suitable conditions, a hyperbolic group with $2$-sphere boundary admits a drilling where the resulting relatively hyperbolic group pair $(\widehat{G}, P)$ has relatively hyperbolic boundary $S^2$. This allows us to reduce the Cannon Conjecture (in the residually finite case) to a relative version, which is likely to be more tractable. |
| title | Drilling hyperbolic groups |
| topic | Geometric Topology Group Theory 20F67, 20F65 |
| url | https://arxiv.org/abs/2406.14667 |