Examples of cubulable groups with fixed-point properties
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914222229684224 |
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| author | Genevois, Anthony |
| author_facet | Genevois, Anthony |
| contents | For every $n \geq 1$, let $(\mathrm{FW}_n)$ denote the fixed-point property for median graphs of cubical dimension $n$ (or equivalently, for CAT(0) cube complexes of dimension $n$). In this article, we construct explicit examples of groups satisfying $(\mathrm{FW}_n)$ but with good cubical properties in higher dimensions. First, we prove that, for a finitely generated group $G$ with no non-abelian free subgroup, $G$ satisfies $(\mathrm{FW}_n)$ if and only if no subgroup $H \leq G$ of index $\leq n$ can be mapped to $\mathbb{D}_\infty$ with an infinite image. For instance, the affine Coxeter group $\tilde{A}_n$ satisfies $(\mathrm{FW}_n)$ but not $(\mathrm{FW}_{n+1})$. In another direction, we investigate virtually graph products of finite groups. As an application of our constructions, we find explicit examples, for every $n \geq 1$, of acylindrically hyperbolic groups that are cocompactly cubulable but satisfy $(\mathrm{FW}_n)$. Several conjectures and open questions are included. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_12402 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Examples of cubulable groups with fixed-point properties Genevois, Anthony Group Theory Metric Geometry 20F65, 20F67, 22D55 For every $n \geq 1$, let $(\mathrm{FW}_n)$ denote the fixed-point property for median graphs of cubical dimension $n$ (or equivalently, for CAT(0) cube complexes of dimension $n$). In this article, we construct explicit examples of groups satisfying $(\mathrm{FW}_n)$ but with good cubical properties in higher dimensions. First, we prove that, for a finitely generated group $G$ with no non-abelian free subgroup, $G$ satisfies $(\mathrm{FW}_n)$ if and only if no subgroup $H \leq G$ of index $\leq n$ can be mapped to $\mathbb{D}_\infty$ with an infinite image. For instance, the affine Coxeter group $\tilde{A}_n$ satisfies $(\mathrm{FW}_n)$ but not $(\mathrm{FW}_{n+1})$. In another direction, we investigate virtually graph products of finite groups. As an application of our constructions, we find explicit examples, for every $n \geq 1$, of acylindrically hyperbolic groups that are cocompactly cubulable but satisfy $(\mathrm{FW}_n)$. Several conjectures and open questions are included. |
| title | Examples of cubulable groups with fixed-point properties |
| topic | Group Theory Metric Geometry 20F65, 20F67, 22D55 |
| url | https://arxiv.org/abs/2311.12402 |