On embeddings of extensions of almost finite actions into cubical shifts
| Autor: | Lanckriet, Emiel, Szabó, Gábor |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | For a countable amenable group $G$ and a fixed dimension $m\geq 1$, we investigate when it is possible to embed a $G$-space $X$ into the $m$-dimensional cubical shift $([0,1]^m)^G$. We focus our attention on systems that arise as an extension of an almost finite $G$-action on a totally disconnected space $Y$, in the sense of Matui and Kerr. We show that if such a $G$-space $X$ has mean dimension less than $m/2$, then $X$ embeds into the $(m+1)$-dimensional cubical shift. If the distinguished factor $G$-space $Y$ is assumed to be a subshift of finite type, then this can be improved to an embedding into the $m$-dimensional cubical shift. This result ought to be viewed as the generalization of a theorem by Gutman-Tsukamoto for $G=\mathbb Z$ to actions of all amenable groups, and represents the first result supporting the Lindenstrauss-Tsukamoto conjecture for actions of groups other than $G=\mathbb{Z}^k$. Comment: 10 pages; v2 minor changes |
| Databáze: | arXiv |
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