Soficity of free extensions of effective subshifts
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909504546799616 |
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| author | Barbieri, Sebastián Sablik, Mathieu Salo, Ville |
| author_facet | Barbieri, Sebastián Sablik, Mathieu Salo, Ville |
| contents | Let $G$ be a group and $H\leqslant G$ a subgroup. The free extension of an $H$-subshift $X$ to $G$ is the $G$-subshift $\widetilde{X}$ whose configurations are those for which the restriction to every coset of $H$ is a configuration from $X$. We study the case of $G = H \times K$ for infinite and finitely generated groups $H$ and $K$: on the one hand we show that if $K$ is nonamenable and $H$ has decidable word problem, then the free extension to $G$ of any $H$-subshift which is effectively closed is a sofic $G$-subshift. On the other hand we prove that if both $H$ and $K$ are amenable, there are always $H$-subshifts which are effectively closed by patterns whose free extension to $G$ is non-sofic. We also present a few applications in the form of a new simulation theorem and a new class of groups which admit strongly aperiodic SFTs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_02620 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Soficity of free extensions of effective subshifts Barbieri, Sebastián Sablik, Mathieu Salo, Ville Dynamical Systems Group Theory 37B10 Let $G$ be a group and $H\leqslant G$ a subgroup. The free extension of an $H$-subshift $X$ to $G$ is the $G$-subshift $\widetilde{X}$ whose configurations are those for which the restriction to every coset of $H$ is a configuration from $X$. We study the case of $G = H \times K$ for infinite and finitely generated groups $H$ and $K$: on the one hand we show that if $K$ is nonamenable and $H$ has decidable word problem, then the free extension to $G$ of any $H$-subshift which is effectively closed is a sofic $G$-subshift. On the other hand we prove that if both $H$ and $K$ are amenable, there are always $H$-subshifts which are effectively closed by patterns whose free extension to $G$ is non-sofic. We also present a few applications in the form of a new simulation theorem and a new class of groups which admit strongly aperiodic SFTs. |
| title | Soficity of free extensions of effective subshifts |
| topic | Dynamical Systems Group Theory 37B10 |
| url | https://arxiv.org/abs/2309.02620 |