Krieger's finite generator theorem for actions of countable groups II

TL;DR

This paper advances the understanding of Rokhlin entropy in group actions, showing that free ergodic actions with finite Rokhlin entropy have nearly Bernoulli generating partitions and establishing key properties of Bernoulli shifts related to entropy and longstanding conjectures.

Contribution

It proves that free ergodic actions with finite Rokhlin entropy have almost Bernoulli generating partitions and links Bernoulli shift entropy to base entropy, addressing major conjectures.

Findings

Every free ergodic action with finite Rokhlin entropy admits almost Bernoulli generating partitions.

The Rokhlin entropy of Bernoulli shifts equals the Shannon entropy of their base.

Bernoulli shifts have completely positive Rokhlin entropy, supporting conjectures.

Abstract

We continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving actions of countable groups introduced in the previous paper. We prove that every free ergodic action with finite Rokhlin entropy admits generating partitions which are almost Bernoulli, strengthening the theorem of Ab\'{e}rt--Weiss that all free actions weakly contain Bernoulli shifts. We then use this result to study the Rokhlin entropy of Bernoulli shifts. Under the assumption that every countable group admits a free ergodic action of positive Rokhlin entropy, we prove that: (i) the Rokhlin entropy of a Bernoulli shift is equal to the Shannon entropy of its base; (ii) Bernoulli shifts have completely positive Rokhlin entropy; and (iii) Gottschalk's surjunctivity conjecture and Kaplansky's direct finiteness conjecture are true.