On embeddings of extensions of almost finite actions into cubical shifts

TL;DR

This paper studies when certain group actions can be embedded into cubical shifts, showing that low mean dimension allows embeddings into higher-dimensional shifts, with improvements under specific conditions, extending previous results to broader groups.

Contribution

It generalizes embedding results from integer actions to all amenable groups, supporting the Lindenstrauss-Tsukamoto conjecture for a wider class of group actions.

Findings

Embedding into (m+1)-dimensional shift if mean dimension < m/2

Improved embedding into m-dimensional shift when the factor is a subshift of finite type

First extension of Gutman-Tsukamoto theorem to all amenable groups

Abstract

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 $G=\mathbb{Z}^k$.