Measure Equivalence Rigidity and Bi-exactness of Groups

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.

34 pages

References in corpus (4)

• Measure equivalence rigidity of the mapping class group
• Free products, Orbit Equivalence and Measure Equivalence Rigidity
• Rigidity results for wreath product II${}_1$ factors
• The class S as an ME invariant

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• Some $OE$ and $W^{\ast }$-rigidity results for actions by wreath product groups
• On fundamental groups of tensor product $\mathrm{I}{\mathrm{I}}_1$ factors
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• Some prime factorization results for free quantum group factors
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