Full groups and soficity

TL;DR

This paper demonstrates that full groups of sofic equivalence relations are sofic, provides a new proof that topological full groups of minimal Cantor homeomorphisms are LEF, and shows certain lamplighter groups are sofic.

Contribution

It proves the soficity of full groups of sofic equivalence relations and offers a simplified proof of LEF property for topological full groups, also establishing soficity for some lamplighter groups.

Findings

Full groups of sofic equivalence relations are sofic.

Topological full groups of minimal Cantor homeomorphisms are LEF.

Certain non-amenable lamplighter groups are sofic.

Abstract

First, we answer a question of Pestov, by proving that the full group of a sofic equivalence relation is a sofic group. Then, we give a short proof of the theorem of Grigorchuk and Medynets that the topological full group of a minimal Cantor homeomorphism is LEF. Finally, we show that for certain non-amenable groups all the generalized lamplighter groups are sofic.