Zariski-Closures of Linear Reflection Groups
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917974255861760 |
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| author | Audibert, Jacques Douba, Sami Lee, Gye-Seon Marquis, Ludovic |
| author_facet | Audibert, Jacques Douba, Sami Lee, Gye-Seon Marquis, Ludovic |
| contents | We give necessary and sufficient conditions for a linear reflection group in the sense of Vinberg to be Zariski-dense in the ambient projective general linear group. As an application, we show that every irreducible right-angled Coxeter group of rank $N \geq 3$ virtually embeds Zariski-densely in $\mathrm{SL}_n(\mathbb{Z})$ for all $n \geq N$. This allows us to settle the existence of Zariski-dense surface subgroups of $\mathrm{SL}_n(\mathbb{Z})$ for all $n \geq 3$. Among the other applications are examples of Zariski-dense one-ended finitely generated subgroups of $\mathrm{SL}_n(\mathbb{Z})$ that are not finitely presented for all $n \geq 6$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_01494 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zariski-Closures of Linear Reflection Groups Audibert, Jacques Douba, Sami Lee, Gye-Seon Marquis, Ludovic Geometric Topology Group Theory 22E40, 20F55 We give necessary and sufficient conditions for a linear reflection group in the sense of Vinberg to be Zariski-dense in the ambient projective general linear group. As an application, we show that every irreducible right-angled Coxeter group of rank $N \geq 3$ virtually embeds Zariski-densely in $\mathrm{SL}_n(\mathbb{Z})$ for all $n \geq N$. This allows us to settle the existence of Zariski-dense surface subgroups of $\mathrm{SL}_n(\mathbb{Z})$ for all $n \geq 3$. Among the other applications are examples of Zariski-dense one-ended finitely generated subgroups of $\mathrm{SL}_n(\mathbb{Z})$ that are not finitely presented for all $n \geq 6$. |
| title | Zariski-Closures of Linear Reflection Groups |
| topic | Geometric Topology Group Theory 22E40, 20F55 |
| url | https://arxiv.org/abs/2504.01494 |