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.
"Classification of the mapping class groups up to measure equivalence." Proc. Japan Acad. Ser. A Math. Sci. 82 (1) 4 - 7, Jan. 2006. https://doi.org/10.3792/pjaa.82.4