Graphs and complexes of lattices
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866917270961258496 |
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| author | Hughes, Sam |
| author_facet | Hughes, Sam |
| contents | We study lattices acting on $\mathrm{CAT}(0)$ spaces via their commensurated subgroups. To do this we introduce the notions of a graph of lattices and a complex of lattices giving graph and complex of group splittings of $\mathrm{CAT}(0)$ lattices. Using this framework we characterise irreducible uniform $(\mathrm{Isom}(\mathbb{E}^n)\times T)$-lattices by $C^\ast$-simplicity and give a necessary condition for lattices in products with a Euclidean factor to be biautomatic. We also construct non-residually finite uniform lattices acting on arbitrary products of right-angled buildings and non-biautomatic lattices acting on the product of $\mathbb{E}^n$ and a right-angled building. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_13728 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Graphs and complexes of lattices Hughes, Sam Group Theory 20F67, 20E08, 57M07, 20E34, 57M60 We study lattices acting on $\mathrm{CAT}(0)$ spaces via their commensurated subgroups. To do this we introduce the notions of a graph of lattices and a complex of lattices giving graph and complex of group splittings of $\mathrm{CAT}(0)$ lattices. Using this framework we characterise irreducible uniform $(\mathrm{Isom}(\mathbb{E}^n)\times T)$-lattices by $C^\ast$-simplicity and give a necessary condition for lattices in products with a Euclidean factor to be biautomatic. We also construct non-residually finite uniform lattices acting on arbitrary products of right-angled buildings and non-biautomatic lattices acting on the product of $\mathbb{E}^n$ and a right-angled building. |
| title | Graphs and complexes of lattices |
| topic | Group Theory 20F67, 20E08, 57M07, 20E34, 57M60 |
| url | https://arxiv.org/abs/2104.13728 |