Mixing and Spectral Gap Relative to Pinsker Factors for Sofic Groups

TL;DR

This paper explores the structural properties of sofic group actions relative to their Pinsker and Outer Pinsker factors, demonstrating mixing and spectral gap properties using advanced representation-theoretic methods.

Contribution

It establishes mixing and spectral gap results for actions of sofic groups relative to their Pinsker and Outer Pinsker factors, extending previous work with novel algebraic techniques.

Findings

Actions are mixing relative to Pinsker and Outer Pinsker factors.

Nonamenable sofic groups have spectral gap relative to these factors.

Uses *-representations of algebraic crossed products instead of unitary representations.

Abstract

Motivated by our previous results, we investigate structural properties of probability measure-preserving actions of sofic groups relative to their Pinsker factor. We also consider the same properties relative to the Outer Pinsker factor, which is another generalization of the Pinsker factor in the nonamenable case. The Outer Pinsker factor is motivated by extension entropy for extensions, which fixes some of the "pathological" behavior of sofic entropy: namely increase of entropy under factor maps. We show that an arbitrary probability measure-preserving action of a sofic group is mixing relative to its Pinsker and Outer Pinsker factors and, if the group is nonamenable, it has spectral gap relative to its Pinsker and Outer Pinsker factors. Our methods are similar to those we developed in "Polish models and sofic entropy" and based on representation-theoretic techniques. One crucial difference is that instead of considering unitary representations of a group we must consider *-representations of algebraic crossed products.