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Measure equivalence rigidity of the mapping class group. (English) Zbl 1277.37005

Summary: We show that the mapping class group of a compact orientable surface with higher complexity satisfies the following rigidity in the sense of measure equivalence: If the mapping class group is measure equivalent to a discrete group, then they are commensurable up to finite kernels. Moreover, we describe all locally compact second countable groups containing a lattice isomorphic to the mapping class group. We obtain similar results for finite direct products of mapping class groups.

MSC:

37A15 General groups of measure-preserving transformations and dynamical systems
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
20F38 Other groups related to topology or analysis
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
22E40 Discrete subgroups of Lie groups
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