Thin hyperbolic reflection groups
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866913198302560256 |
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| author | Bogachev, Nikolay Kolpakov, Alexander |
| author_facet | Bogachev, Nikolay Kolpakov, Alexander |
| contents | We study a family of Zariski dense finitely generated discrete subgroups of $\mathrm{Isom}(\mathbb{H}^d)$, $d \geqslant 2$, defined by the following property: any group in this family contains at least one reflection in a hyperplane. As an application we obtain a general description of all thin hyperbolic reflection groups. In particular, we show that the Vinberg algorithm applied to a non-reflective Lorentzian lattice gives rise to an infinite sequence of thin reflection subgroups in $\mathrm{Isom}(\mathbb{H}^d)$, for any $d \geqslant 2$. Moreover, every such group is a subgroup of a group produced by the Vinberg algorithm applied to a Lorentzian lattice independently on the latter being reflective. As a consequence, all thin hyperbolic reflection groups are enumerable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_14642 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Thin hyperbolic reflection groups Bogachev, Nikolay Kolpakov, Alexander Group Theory Geometric Topology Number Theory 22E40, 20F55 We study a family of Zariski dense finitely generated discrete subgroups of $\mathrm{Isom}(\mathbb{H}^d)$, $d \geqslant 2$, defined by the following property: any group in this family contains at least one reflection in a hyperplane. As an application we obtain a general description of all thin hyperbolic reflection groups. In particular, we show that the Vinberg algorithm applied to a non-reflective Lorentzian lattice gives rise to an infinite sequence of thin reflection subgroups in $\mathrm{Isom}(\mathbb{H}^d)$, for any $d \geqslant 2$. Moreover, every such group is a subgroup of a group produced by the Vinberg algorithm applied to a Lorentzian lattice independently on the latter being reflective. As a consequence, all thin hyperbolic reflection groups are enumerable. |
| title | Thin hyperbolic reflection groups |
| topic | Group Theory Geometric Topology Number Theory 22E40, 20F55 |
| url | https://arxiv.org/abs/2112.14642 |