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 non-amenable 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 non-amenable exact direct product groups.