Measure equivalence for non-unimodular groups

TL;DR

This paper explores measure equivalence among locally compact, second countable groups, offering new operator algebraic and ergodic theoretic perspectives, and classifies amenable groups within this framework.

Contribution

It provides a comprehensive classification of amenable groups up to measure equivalence and introduces reformulations using operator algebra and ergodic theory.

Findings

Complete classification of amenable groups up to measure equivalence

Operator algebraic reformulations of measure equivalence

Ergodic theoretic reformulations of measure equivalence

Abstract

We undertake a comprehensive study of measure equivalence between general locally compact, second countable groups, providing operator algebraic and ergodic theoretic reformulations, and complete the classification of amenable groups within this class up to measure equivalence.