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Classification of the mapping class groups up to measure equivalence. (English) Zbl 1130.37309

Summary: We study the mapping class groups of compact orientable surfaces from the viewpoint of measure equivalence. In this paper, we announce some classification result of the mapping class groups in terms of measure equivalence and the result that there exist various kinds of discrete groups which are not measure equivalent to the mapping class groups. As a byproduct of the proof, it turns out that the mapping class group is exact.

MSC:

37A15 General groups of measure-preserving transformations and dynamical systems
20F28 Automorphism groups of groups
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
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