Approximate homomorphisms and sofic approximations of orbit equivalence relations

TL;DR

This paper demonstrates that approximate homomorphisms of countable groups can be realized as restrictions of sofic approximations of orbit equivalence relations, linking group invariants to dynamical structures.

Contribution

It establishes a novel connection between approximate homomorphisms and sofic approximations of orbit equivalence relations, highlighting the role of invariant random subgroups.

Findings

Approximate homomorphisms can be realized as restrictions of sofic approximations.

Orbit equivalence relations are uniquely determined by invariant random subgroups.

Applications recover known stability and conjugacy results for amenable groups.

Abstract

We show that for every countable group, any sequence of approximate homomorphisms with values in permutations can be realized as the restriction of a sofic approximation of an orbit equivalence relation. Moreover, this orbit equivalence relation is uniquely determined by the invariant random subgroup of the approximate homomorphism. We record applications of this result to recover various known stability and conjugacy characterizations for almost homomorphisms of amenable groups.