Ping-pong in the projective plane over a nonarchimedean field
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909616875503616 |
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| author | Douba, Sami Kubrak, Dmitry Tsouvalas, Konstantinos |
| author_facet | Douba, Sami Kubrak, Dmitry Tsouvalas, Konstantinos |
| contents | We show that any lattice in $\mathrm{SL}_3(k)$, where $k$ is a nonarchimedean local field, contains an undistorted subgroup isomorphic to the free product $\mathbb{Z}^2*\mathbb{Z}$. To our knowledge, the subgroups we construct give the first examples in the literature of finitely generated Zariski-dense infinite-covolume discrete subgroups of an almost simple group over a nonarchimedean local field that are not virtually free. Our result is in contrast to the case of $\mathrm{SL}_3(\mathbb{Z})$, in which the existence of a $\mathbb{Z}^2*\mathbb{Z}$ subgroup remains open. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_13639 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ping-pong in the projective plane over a nonarchimedean field Douba, Sami Kubrak, Dmitry Tsouvalas, Konstantinos Group Theory Number Theory We show that any lattice in $\mathrm{SL}_3(k)$, where $k$ is a nonarchimedean local field, contains an undistorted subgroup isomorphic to the free product $\mathbb{Z}^2*\mathbb{Z}$. To our knowledge, the subgroups we construct give the first examples in the literature of finitely generated Zariski-dense infinite-covolume discrete subgroups of an almost simple group over a nonarchimedean local field that are not virtually free. Our result is in contrast to the case of $\mathrm{SL}_3(\mathbb{Z})$, in which the existence of a $\mathbb{Z}^2*\mathbb{Z}$ subgroup remains open. |
| title | Ping-pong in the projective plane over a nonarchimedean field |
| topic | Group Theory Number Theory |
| url | https://arxiv.org/abs/2505.13639 |