Entropy, Shannon orbit equivalence, and sparse connectivity

TL;DR

This paper introduces Shannon orbit equivalence for group actions, showing that under certain conditions, it preserves maximum sofic entropy and characterizes Bernoulli actions, extending previous results to broader classes.

Contribution

It extends the concept of Shannon orbit equivalence to non-finitely generated groups with specific properties, establishing entropy invariance and characterizations of Bernoulli actions.

Findings

Shannon orbit equivalence implies equal maximum sofic entropy for certain groups.

Bernoulli actions are Shannon orbit equivalent if and only if they are measure conjugate.

The results apply to actions with a new sparse connectivity property called SC.

Abstract

We say that two free probability-measure-preserving actions of countable groups are Shannon orbit equivalent if there is an orbit equivalence between them whose associated cocycle partitions have finite Shannon entropy. We show that if the acting groups are sofic and each has a w-normal amenable subgroup which is neither locally finite nor virtually cyclic then Shannon orbit equivalence implies that the actions have the same maximum sofic entropy. This extends a result of Austin beyond the finitely generated amenable setting and has the consequence that two Bernoulli actions of a group with the properties in question are Shannon orbit equivalent if and only if they are measure conjugate. Our arguments apply more generally to actions satisfying a sparse connectivity condition which we call property SC, and yield an entropy inequality under the assumption that one of the actions has this property.