Orbit equivalence rigidity and bounded cohomology


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.


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