Convex cocompactness for Coxeter groups
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866917770886643712 |
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| author | Danciger, Jeffrey Guéritaud, François Kassel, Fanny Lee, Gye-Seon Marquis, Ludovic |
| author_facet | Danciger, Jeffrey Guéritaud, François Kassel, Fanny Lee, Gye-Seon Marquis, Ludovic |
| contents | We investigate representations of Coxeter groups into $\mathrm{GL}(n,\mathbb{R})$ as geometric reflection groups which are convex cocompact in the projective space $\mathbb{P}(\mathbb{R}^n)$. We characterize which Coxeter groups admit such representations, and we fully describe the corresponding spaces of convex cocompact representations as reflection groups, in terms of the associated Cartan matrices. The Coxeter groups that appear include all infinite, word hyperbolic Coxeter groups; for such groups the representations as reflection groups that we describe are exactly the projective Anosov ones. We also obtain a large class of nonhyperbolic Coxeter groups, thus providing many examples for the theory of nonhyperbolic convex cocompact subgroups in $\mathbb{P}(\mathbb{R}^n)$ developed in arXiv:1704.08711. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_02757 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Convex cocompactness for Coxeter groups Danciger, Jeffrey Guéritaud, François Kassel, Fanny Lee, Gye-Seon Marquis, Ludovic Group Theory Geometric Topology We investigate representations of Coxeter groups into $\mathrm{GL}(n,\mathbb{R})$ as geometric reflection groups which are convex cocompact in the projective space $\mathbb{P}(\mathbb{R}^n)$. We characterize which Coxeter groups admit such representations, and we fully describe the corresponding spaces of convex cocompact representations as reflection groups, in terms of the associated Cartan matrices. The Coxeter groups that appear include all infinite, word hyperbolic Coxeter groups; for such groups the representations as reflection groups that we describe are exactly the projective Anosov ones. We also obtain a large class of nonhyperbolic Coxeter groups, thus providing many examples for the theory of nonhyperbolic convex cocompact subgroups in $\mathbb{P}(\mathbb{R}^n)$ developed in arXiv:1704.08711. |
| title | Convex cocompactness for Coxeter groups |
| topic | Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2102.02757 |