  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.
