Soficity, Amenability, and LEF-ness for topological full groups

TL;DR

This paper investigates the approximation properties of topological full groups of Cantor set actions, establishing conditions for amenability, LEF-ness, and soficity, with implications for dynamical systems and group theory.

Contribution

It introduces new methods to detect hyperfiniteness in sofic approximations and extends LEF-ness results to broader classes of topological full groups.

Findings

Amenability of topological full groups is equivalent to the group's amenability.

Topological full groups of zero entropy actions are generally non-amenable.

Certain Toeplitz subshifts' topological full groups are LEF and sofic.

Abstract

In this paper, we study several finite approximation properties of topological full groups of group actions on the Cantor set such that free points are dense. Firstly, we establish that for such a distal action $\alpha$ of a countable discrete group $G$ on the Cantor set, the topological full group $[[\alpha]]$ is amenable if and only if $G$ is amenable. This result is obtained through a novel method that detects hyperfiniteness in certain sofic approximation graph sequences of finitely generated subgroups of $[[\alpha]]$. We also provide estimates for related F{\o}lner functions. Next, we obtain negative results on the amenability of topological full groups for actions with zero topological entropy by calculating the topological entropy of certain examples provided by Elek and Monod. Furthermore, we demonstrate that the topological full group $[[\alpha]]$ of a minimal topologically free residually finite action $\alpha$ on the Cantor set is locally embeddable in the class of finite groups (LEF). This generalizes a result previously obtained by Grigorchuk and Medynets in the case of minimal $\mathbb{Z}$-actions. As an application, we show that topological full groups of certain Toeplitz subshifts on free groups are LEF and therefore sofic.