Measure conjugacy invariants for actions of countable sofic groups

TL;DR

This paper introduces measure-conjugacy invariants for actions of countable sofic groups, generalizing entropy, and applies them to classify Bernoulli shifts and other systems up to conjugacy and equivalence.

Contribution

It defines new invariants for sofic group actions, extending entropy concepts, and achieves classification results for Bernoulli shifts over various complex groups.

Findings

Invariants computed explicitly for Bernoulli shifts over sofic groups

Complete classification of Bernoulli systems up to measure-conjugacy for many groups

Applications to orbit and von Neumann equivalence classifications

Abstract

Sofic groups were defined implicitly by Gromov in [Gr99] and explicitly by Weiss in [We00]. All residually finite groups (and hence every linear group) is sofic. The purpose of this paper is to introduce, for every countable sofic group $G$, a family of measure-conjugacy invariants for measure-preserving $G$-actions on probability spaces. These invariants generalize Kolmogorov-Sinai entropy for actions of amenable groups. They are computed exactly for Bernoulli shifts over $G$, leading to a complete classification of Bernoulli systems up to measure-conjugacy for many groups including all countable linear groups. Recent rigidity results of Y. Kida and S. Popa are utilized to classify Bernoulli shifts over mapping class groups and property T groups up to orbit equivalence and von Neumann equivalence respectively.