The geometry of model spaces for probability-preserving actions of sofic groups

TL;DR

This paper investigates the geometric properties of model spaces associated with sofic entropy in group actions, providing new insights into their structure and implications for Bernoulli shift factors.

Contribution

It introduces an approximate connectedness property of model spaces and offers a new proof regarding non-Bernoulli factors of Bernoulli shifts for certain groups.

Findings

Model spaces can exhibit approximate connectedness.

The new proof applies to fewer groups but offers additional insights.

Provides a geometric perspective on entropy invariants.

Abstract

Bowen's notion of sofic entropy is a powerful invariant for classifying probability-preserving actions of sofic groups. It can be defined in terms of the covering numbers of certain metric spaces associated to such an action, the `model spaces'. The metric geometry of these model spaces can exhibit various interesting features, some of which provide other invariants of the action. This paper explores an approximate connectedness property of the model spaces, and uses it give a new proof that certain groups admit factors of Bernoulli shifts which are not Bernoulli. This was originally proved by Popa. Our proof covers fewer examples than his, but provides additional information about this phenomenon.