On Amenability and LEF-ness for topological full groups

Abstract

In this paper, we prove that for a topologically free 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 generalizes a result by Cortez and Medynets. In addition, 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 extends 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.

This is the accepted version by Journal of Algebra. The title is changed and the proof of Theorem A has been simplified