Sofic homological invariants and the Weak Pinsker Property

TL;DR

This paper constructs a counterexample to the Weak Pinsker Property for actions of non-abelian free groups using sofic entropy, introducing new homology-based invariants to distinguish actions.

Contribution

It introduces homology growth invariants for measure conjugacy and demonstrates a counterexample to the WPP in non-amenable group actions.

Findings

Counterexample with exponential homology growth in dimension 0.

Actions with WPP have subexponential homology growth.

Sofic entropy-based methods distinguish different group actions.

Abstract

A probability-measure-preserving transformation has the Weak Pinsker Property (WPP) if for every $\epsilon>0$ it is measurably conjugate to the direct product of a transformation with entropy $<\epsilon$ and a Bernoulli shift. In a recent breakthrough, Tim Austin proved that every ergodic transformation satisfies this property. Moreover, the natural analog for amenable group actions is also true. By contrast, this paper provides a counterexample in which the group $\Gamma$ is a non-abelian free group and the notion of entropy is sofic entropy. The counterexample is a limit of hardcore models on random regular graphs. In order to prove that it does not have the WPP, this paper introduces new measure conjugacy invariants based on the growth of homology of the model spaces of the action. The main result is obtained by showing that any action with the WPP has subexponential homology growth in dimension $0$, while the counterexample has exponential homology growth in dimension $0$.