# Orbit equivalence rigidity and bounded cohomology

### Abstract

We establish new results and introduce new methods in the theory of measurable orbit equivalence, using bounded cohomology of group representations. Our rigidity statements hold for a wide (uncountable) class of groups arising from negative curvature geometry. Amongst our applications are (a) measurable Mostow-type rigidity theorems for products of negatively curved groups; (b) prime factorization results for measure equivalence; (c) superrigidity for orbit equivalence; (d) the first examples of continua of type $II_1$ equivalence relations with trivial outer automorphism group that are mutually not stably isomorphic.

## Authors

Nicolas Monod

Department of Mathematics
University of Chicago
Chicago, IL 60637
United States

Yehuda Shalom

School of Mathematical Sciences
Tel Aviv University
Tel Aviv 69978
Israel