A survey of Measured Group Theory

TL;DR

This survey reviews recent advances in measured group theory, focusing on measure and orbit equivalence, invariants, and rigidity phenomena in the classification of infinite groups and their actions.

Contribution

It provides a comprehensive overview of measure and orbit equivalence, highlighting key invariants and rigidity results in the classification of groups and their actions.

Findings

Summary of measure equivalence invariants

Overview of orbit equivalence classification results

Discussion of rigidity phenomena in measured group theory

Abstract

The title refers to the area of research which studies infinite groups using measure-theoretic tools, and studies the restrictions that group structure imposes on ergodic theory of their actions. The paper is a survey of recent developments focused on the notion of Measure Equivalence between groups, and Orbit Equivalence between group actions. We discuss known invariants and classification results (rigidity) in both areas.