tailieunhanh - Đề tài " 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 II1 equivalence relations with trivial outer automorphism group that are mutually not stably isomorphic. Contents 1. Introduction 2. Discussion and applications of the main results 3. Background. | Annals of Mathematics Orbit equivalence rigidity and bounded cohomology By Nicolas Monod and Yehuda Shalom Annals of Mathematics 164 2006 825 878 Orbit equivalence rigidity and bounded cohomology By Nicolas Monod and Yehuda Shalom 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 II1 equivalence relations with trivial outer automorphism group that are mutually not stably isomorphic. Contents 1. Introduction 2. Discussion and applications of the main results 3. Background in bounded cohomology 4. Cohomological induction through couplings 5. Strong rigidity 6. Superrigidity 7. Groups in the class C and ME invariants References 1. Introduction In this paper a companion to MS2 we continue our attempts to widen the scope of rigidity theory using new techniques made available by the bounded cohomology methods recently developed by Burger and Monod BM2 M . In the present paper we focus our attention on rigidity of measurable orbit equivalence an area which has seen remarkable achievements by R. Zimmer during the 80 s and in the last few years has flourished again with the striking work of A. Furman F1 F2 F3 and D. Gaboriau Ga1 Ga2 Ga3 . Our main purpose is to establish new rigidity phenomena some reminiscent of those known in the case of higher rank lattices for a large uncountable class of groups arising geometrically in the general framework of negative curvature 826 NICOLAS MONOD AND YEHUDA SHALOM Examples . Consider the collection of all countable groups r which admit either i A nonelementary simplicial action

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