Model geometries of finitely generated groups
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866917769629401088 |
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| author | Margolis, Alex |
| author_facet | Margolis, Alex |
| contents | We study model geometries of finitely generated groups. If a finitely generated group does not contain a non-trivial finite rank free abelian commensurated subgroup, we show any model geometry is dominated by either a symmetric space of non-compact type, an infinite locally finite vertex-transitive graph, or a product of such spaces. We also prove that a finitely generated group possesses a model geometry not dominated by a locally finite graph if and only if it contains either a commensurated finite rank free abelian subgroup, or a uniformly commensurated subgroup that is a uniform lattice in a semisimple Lie group. This characterises finitely generated groups that embed as uniform lattices in locally compact groups that are not compact-by-(totally disconnected). We show the only such groups of cohomological two are surface groups and generalised Baumslag-Solitar groups, and we obtain an analogous characterisation for groups of cohomological dimension three. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_10509 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Model geometries of finitely generated groups Margolis, Alex Group Theory 20F65 (Primary) 20F67 57M07 (Secondary) We study model geometries of finitely generated groups. If a finitely generated group does not contain a non-trivial finite rank free abelian commensurated subgroup, we show any model geometry is dominated by either a symmetric space of non-compact type, an infinite locally finite vertex-transitive graph, or a product of such spaces. We also prove that a finitely generated group possesses a model geometry not dominated by a locally finite graph if and only if it contains either a commensurated finite rank free abelian subgroup, or a uniformly commensurated subgroup that is a uniform lattice in a semisimple Lie group. This characterises finitely generated groups that embed as uniform lattices in locally compact groups that are not compact-by-(totally disconnected). We show the only such groups of cohomological two are surface groups and generalised Baumslag-Solitar groups, and we obtain an analogous characterisation for groups of cohomological dimension three. |
| title | Model geometries of finitely generated groups |
| topic | Group Theory 20F65 (Primary) 20F67 57M07 (Secondary) |
| url | https://arxiv.org/abs/2207.10509 |