Subgroups of CAT(0) groups, exotic finiteness properties and non-QI-embeddings into linear groups
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913617160437760 |
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| author | Isenrich, Claudio Llosa Tsouvalas, Konstantinos |
| author_facet | Isenrich, Claudio Llosa Tsouvalas, Konstantinos |
| contents | For every positive integer $n$ we construct an example of a subgroup $L< G$ of a linear ${\rm CAT}(0)$ group $G$ such that $L$ is of finiteness type $\mathcal{F}_{n-1}$ and not $\mathcal{F}_n$, and $L$ does not admit a representation into $\mathsf{GL}_d(k)$ which is a quasi-isometric embedding for any local field $k$. We further prove that there is a faithful representation of $L$ into some $\mathsf{GL}_{\ell}(\mathbb{C})$ which is not the restriction of any representation of $G$. This generalises a family of fibre products of type $\mathcal{F}_2$ not $\mathcal{F}_3$ with these properties constructed by the second author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_14081 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Subgroups of CAT(0) groups, exotic finiteness properties and non-QI-embeddings into linear groups Isenrich, Claudio Llosa Tsouvalas, Konstantinos Group Theory Geometric Topology 20F67, 20F65, 22E40 For every positive integer $n$ we construct an example of a subgroup $L< G$ of a linear ${\rm CAT}(0)$ group $G$ such that $L$ is of finiteness type $\mathcal{F}_{n-1}$ and not $\mathcal{F}_n$, and $L$ does not admit a representation into $\mathsf{GL}_d(k)$ which is a quasi-isometric embedding for any local field $k$. We further prove that there is a faithful representation of $L$ into some $\mathsf{GL}_{\ell}(\mathbb{C})$ which is not the restriction of any representation of $G$. This generalises a family of fibre products of type $\mathcal{F}_2$ not $\mathcal{F}_3$ with these properties constructed by the second author. |
| title | Subgroups of CAT(0) groups, exotic finiteness properties and non-QI-embeddings into linear groups |
| topic | Group Theory Geometric Topology 20F67, 20F65, 22E40 |
| url | https://arxiv.org/abs/2412.14081 |