On Sofic Actions and Equivalence Relations

TL;DR

This paper extends the concept of sofic equivalence relations into an operator algebraic framework, proves the equivalence of different definitions, and shows that certain group and action constructions preserve soficity.

Contribution

It introduces a new operator algebraic definition of sofic equivalence relations and demonstrates that amalgamated products of sofic groups and actions over amenable groups remain sofic.

Findings

Equivalence of graph-theoretic and operator algebraic definitions of sofic relations.

Amalgamated product of sofic actions over amenable groups is sofic.

Amalgamated product of sofic groups over an amenable subgroup is sofic.

Abstract

The notion of sofic equivalence relation was introduced by Gabor Elek and Gabor Lippner. Their technics employ some graph theory. Here we define this notion in a more operator algebraic context, starting from Connes' embedding problem, and prove the equivalence of this two definitions. We introduce a notion of sofic action for an arbitrary group and prove that amalgamated product of sofic actions over amenable groups is again sofic. We also prove that amalgamated product of sofic groups over an amenable subgroup is again sofic.