The mapping class group from the viewpoint of measure equivalence theory

Details The mapping class group from the viewpoint of measure equivalence theory The mapping class group from the viewpoint of measure equivalence theory

2005-12-12

arxiv.org/abs/math/0512230v2

Mem.Amer.Math.Soc. 196 (2008)

Mathematics

Group Theory

vi+190 pages, 11 figures

Scientific paper

We obtain some classification result for the mapping class groups of compact orientable surfaces in terms of measure equivalence. In particular, the mapping class groups of different closed surfaces can not be measure equivalent. Moreover, we give various examples of discrete groups which are not measure equivalent to the mapping class groups. In the course of the proof, we investigate amenability in a measurable sense for the actions of the mapping class group on the boundary at infinity of the curve complex and on the Thurston boundary. Using this investigation, we prove that the mapping class group of a compact orientable surface is exact.

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