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Orbit equivalence rigidity and bounded cohomology. (English) Zbl 1129.37003

Summary: 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.

MSC:

37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
22F10 Measurable group actions
37A15 General groups of measure-preserving transformations and dynamical systems
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