Sofic representations of amenable groups

TL;DR

This paper proves that the free product of two sofic groups over any amenable subgroup remains sofic, and characterizes amenable groups by their unique sofic representation up to conjugacy.

Contribution

It generalizes previous results by removing the monotileability restriction and establishes a new characterization of amenable groups via sofic representations.

Findings

The free product of two sofic groups over an arbitrary amenable subgroup is sofic.

A finitely generated group is amenable if and only if it has a unique sofic representation up to conjugacy.

The result extends the class of groups known to be sofic and provides a new perspective on amenability.

Abstract

Using probabilistic methods, Collins and Dykema proved that the free product of two sofic groups amalgamated over a monotileably amenable subgroup is sofic as well. We show that the restriction is unnecessary; the free product of two sofic groups amalgamated over an arbitrary amenable subgroup is sofic. We also prove a group theoretical analogue of a result of Kenley Jung. A finitely generated group is amenable if and only if it has only one sofic representation up to conjugacy equivalence.