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

Xin Ma

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 $α$ of a countable discrete group $G$ on the Cantor set, the topological full group $[[α]]$ 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 $[[α]]$. We also provide estimates for related Følner 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 $[[α]]$ of a minimal topologically free residually finite action $α$ 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.

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

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 of a countable discrete group on the Cantor set, the topological full group is amenable if and only if is amenable. This result is obtained through a novel method that detects hyperfiniteness in certain sofic approximation graph sequences of finitely generated subgroups of . We also provide estimates for related Følner 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 of a minimal topologically free residually finite action 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 -actions. As an application, we show that topological full groups of certain Toeplitz subshifts on free groups are LEF and therefore sofic.
Paper Structure (8 sections, 33 theorems, 109 equations)

This paper contains 8 sections, 33 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Soficity of topological full groups
  3. Amenability of topological full groups
  4. Hyperfiniteness of graph sequences
  5. Sofic approximations and amenability for topological full groups
  6. Non-amenability of topological full groups
  7. Residually finite actions and LEF topological full groups
  8. Acknowledgement

Key Result

Theorem A

Let $\alpha: G\curvearrowright X$ be a distal action of an infinite countable discrete group $G$ on the Cantor set $X$ such that free points are dense. Then $[[\alpha]]$ is amenable if and only if $G$ is amenable.

Theorems & Definitions (72)

  • Theorem A: Theorem \ref{['thm: equi-amenable']}
  • Corollary 1.2
  • Theorem B: Theorem \ref{['thm: non-amenable']}
  • Theorem C: Theorem \ref{['thm: topo group of residually finite action']}
  • Corollary 1.3
  • Definition 2.1: El
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • ...and 62 more