Hyperfinite measure-preserving actions of countable groups and their model theory

TL;DR

This paper provides a concise proof of a key theorem relating hyperfinite measure-preserving actions of countable groups to their invariant random subgroups and explores their model theory, extending previous automorphism studies.

Contribution

It offers a shorter proof of Elek's theorem and applies it to advance the model theory of hyperfinite actions, generalizing prior automorphism research.

Findings

Two hyperfinite actions are approximately conjugate iff they share the same invariant random subgroup.

The work extends the model-theoretic framework to hyperfinite measure-preserving actions.

Provides new insights into the structure and classification of such actions.

Abstract

We give a shorter proof of a theorem of G. Elek stating that two hyperfinite measure-preserving actions of a countable group on standard probability spaces are approximately conjugate if and only if they have the same invariant random subgroup. We then use this theorem to study model theory of hyperfinite measure-preserving actions of countable groups on probability spaces. This work generalizes the model-theoretic study of automorphisms of probability spaces conducted by I. Ben Yaacov, A. Berenstein, C. W. Henson and A. Usvyatsov.