Measure equivalence rigidity and bi-exactness of groups

We get three types of results on measurable group theory; direct product groups of Ozawa’s class S groups, wreath product groups and amalgamated free products.We prove measure equivalence factorization results on direct product groups of Ozawa’s class S groups. As consequences, Monod–Shalom type orbit equivalence rigidity theorems follow. We prove that if two wreath product groups A G, B Γ of nonamenable exact direct product groups G, Γ with amenable bases A, B are measure equivalent, then G and Γ are measure equivalent. We get Bass–Serre rigidity results on amalgamated free products of nonamenable exact direct product groups. © 2009 Elsevier Inc. All rights reserved.