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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:
37A15General groups of measure-preserving transformation
37A20Orbit equivalence, cocycles, ergodic equivalence relations
20F38Other groups related to topology or analysis
37A35Entropy and other invariants, isomorphism, classification (ergodic theory)
37A15General groups of measure-preserving transformation
22E40Discrete subgroups of Lie groups