Measure Equivalence Rigidity and Bi-exactness of Groups

TL;DR

This paper establishes measure equivalence rigidity results for complex group constructions like direct products, wreath products, and amalgamated free products, revealing deep structural invariances under measure equivalence.

Contribution

It introduces new measure equivalence factorization and rigidity theorems for groups in Ozawa's class S, wreath products, and amalgamated free products, expanding understanding of their structural properties.

Findings

Measure equivalence factorization for direct product groups of class S

Orbit equivalence rigidity for certain product groups

Bass-Serre rigidity results for amalgamated free products

Abstract

We get three types of results on measure equivalence rigidity; direct product groups of Ozawa's class $\mathcal{S}$ groups, wreath product groups and amalgamated free products. We prove measure equivalence factorization results on direct product groups of Ozawa's class $\mathcal{S}$ groups. As consequences, Monod--Shalom type orbit equivalence rigidity theorems follow. We prove that if two wreath product groups $A \wr G$, $B \wr \Gamma$ of non-amenable exact direct product groups $G$, $\Gamma$ with amenable bases $A$, $B$ are measure equivalent, then $G$ and $\Gamma$ are measure equivalent. We get Bass--Serre rigidity results on amalgamated free products of non-amenable exact direct product groups.