Self-simulable groups
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866912242695405568 |
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| author | Barbieri, Sebastián Sablik, Mathieu Salo, Ville |
| author_facet | Barbieri, Sebastián Sablik, Mathieu Salo, Ville |
| contents | We say that a finitely generated group $Γ$ is self-simulable if every effectively closed action of $Γ$ on a closed subset of $\{\texttt{0},\texttt{1}\}^{\mathbb{N}}$ is the topological factor of a $Γ$-subshift of finite type. We show that self-simulable groups exist, that any direct product of non-amenable finitely generated groups is self-simulable, that under technical conditions self-simulability is inherited from subgroups, and that the subclass of self-simulable groups is stable under commensurability and quasi-isometries of finitely presented groups.
Some notable examples of self-simulable groups obtained are the direct product $F_k \times F_k$ of two free groups of rank $k \geq 2$, non-amenable finitely generated branch groups, the simple groups of Burger and Mozes, Thompson's $V$, the groups $\operatorname{GL}_n(\mathbb{Z})$, $\operatorname{SL}_n(\mathbb{Z})$, $\operatorname{Aut}(F_n)$ and $\operatorname{Out}(F_n)$ for $n \geq 5$; The braid groups $B_m$ for $m \geq 7$, and certain classes of RAAGs. We also show that Thompson's $F$ is self-simulable if and only if $F$ is non-amenable, thus giving a computability characterization of this well-known open problem. We also exhibit a few applications of self-simulability on the dynamics of these groups, notably, that every self-simulable group with decidable word problem admits a nonempty strongly aperiodic subshift of finite type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_05141 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Self-simulable groups Barbieri, Sebastián Sablik, Mathieu Salo, Ville Group Theory Dynamical Systems Primary: 37B10. Secondary: 37B05, 20F10 We say that a finitely generated group $Γ$ is self-simulable if every effectively closed action of $Γ$ on a closed subset of $\{\texttt{0},\texttt{1}\}^{\mathbb{N}}$ is the topological factor of a $Γ$-subshift of finite type. We show that self-simulable groups exist, that any direct product of non-amenable finitely generated groups is self-simulable, that under technical conditions self-simulability is inherited from subgroups, and that the subclass of self-simulable groups is stable under commensurability and quasi-isometries of finitely presented groups. Some notable examples of self-simulable groups obtained are the direct product $F_k \times F_k$ of two free groups of rank $k \geq 2$, non-amenable finitely generated branch groups, the simple groups of Burger and Mozes, Thompson's $V$, the groups $\operatorname{GL}_n(\mathbb{Z})$, $\operatorname{SL}_n(\mathbb{Z})$, $\operatorname{Aut}(F_n)$ and $\operatorname{Out}(F_n)$ for $n \geq 5$; The braid groups $B_m$ for $m \geq 7$, and certain classes of RAAGs. We also show that Thompson's $F$ is self-simulable if and only if $F$ is non-amenable, thus giving a computability characterization of this well-known open problem. We also exhibit a few applications of self-simulability on the dynamics of these groups, notably, that every self-simulable group with decidable word problem admits a nonempty strongly aperiodic subshift of finite type. |
| title | Self-simulable groups |
| topic | Group Theory Dynamical Systems Primary: 37B10. Secondary: 37B05, 20F10 |
| url | https://arxiv.org/abs/2104.05141 |