Presentations for Singly-Cusped Bianchi Groups
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866913989121802240 |
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| author | Reese, Tanner |
| author_facet | Reese, Tanner |
| contents | We produce a complete list of group presentations for singly-cusped Bianchi groups, $PSL_2 (\mathcal{O}_d)$ where $\mathcal{O}_d$ is the ring of integers for $\mathbb{Q}(\sqrt{d})$ and $d$ is -1, -2, -3, -7, -11, -19, -43, -67, or -163. To do this, we apply a theorem due to Macbeath to a sufficiently large horoball, treating the Bianchi groups as discrete subgroups of isometries for $\mathbf{H}^3$. As far as we know, explicit presentations were not previously known when $d$ is -43, -67, or -163. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_01262 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Presentations for Singly-Cusped Bianchi Groups Reese, Tanner Group Theory Geometric Topology We produce a complete list of group presentations for singly-cusped Bianchi groups, $PSL_2 (\mathcal{O}_d)$ where $\mathcal{O}_d$ is the ring of integers for $\mathbb{Q}(\sqrt{d})$ and $d$ is -1, -2, -3, -7, -11, -19, -43, -67, or -163. To do this, we apply a theorem due to Macbeath to a sufficiently large horoball, treating the Bianchi groups as discrete subgroups of isometries for $\mathbf{H}^3$. As far as we know, explicit presentations were not previously known when $d$ is -43, -67, or -163. |
| title | Presentations for Singly-Cusped Bianchi Groups |
| topic | Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2206.01262 |