Strong topological Rokhlin property, shadowing, and symbolic dynamics of countable groups
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911791099936768 |
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| author | Doucha, Michal |
| author_facet | Doucha, Michal |
| contents | A countable group $G$ has the strong topological Rokhlin property (STRP) if it admits a continuous action on the Cantor space with a comeager conjugacy class. We show that having the STRP is a symbolic dynamical property. We prove that a countable group $G$ has the STRP if and only if certain sofic subshifts over $G$ are dense in the space of subshifts. A sufficient condition is that isolated shifts over $G$ are dense in the space of all subshifts.
We provide numerous applications including the proof that a group that decomposes as a free product of finite or cyclic groups has the STRP. We show that finitely generated nilpotent groups do not have the STRP unless they are virtually cyclic; the same is true for many groups of the form $G_1\times G_2\times G_3$ where each factor is recursively presented. We show that a large class of non-finitely generated groups do not have the STRP, this includes any group with infinitely generated center and the Hall universal locally finite group.
We find a very strong connection between the STRP and shadowing, a.k.a. pseudo-orbit tracing property. We show that shadowing is generic for actions of a finitely generated group $G$ if and only if $G$ has the STRP. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_08145 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Strong topological Rokhlin property, shadowing, and symbolic dynamics of countable groups Doucha, Michal Dynamical Systems Group Theory A countable group $G$ has the strong topological Rokhlin property (STRP) if it admits a continuous action on the Cantor space with a comeager conjugacy class. We show that having the STRP is a symbolic dynamical property. We prove that a countable group $G$ has the STRP if and only if certain sofic subshifts over $G$ are dense in the space of subshifts. A sufficient condition is that isolated shifts over $G$ are dense in the space of all subshifts. We provide numerous applications including the proof that a group that decomposes as a free product of finite or cyclic groups has the STRP. We show that finitely generated nilpotent groups do not have the STRP unless they are virtually cyclic; the same is true for many groups of the form $G_1\times G_2\times G_3$ where each factor is recursively presented. We show that a large class of non-finitely generated groups do not have the STRP, this includes any group with infinitely generated center and the Hall universal locally finite group. We find a very strong connection between the STRP and shadowing, a.k.a. pseudo-orbit tracing property. We show that shadowing is generic for actions of a finitely generated group $G$ if and only if $G$ has the STRP. |
| title | Strong topological Rokhlin property, shadowing, and symbolic dynamics of countable groups |
| topic | Dynamical Systems Group Theory |
| url | https://arxiv.org/abs/2211.08145 |