Memoirs of the American Mathematical Society 2008; 190 pp; softcover Volume: 196 ISBN10: 0821841963 ISBN13: 9780821841969 List Price: US$80 Individual Members: US$48 Institutional Members: US$64 Order Code: MEMO/196/916
 The author obtains 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 cannot be measure equivalent. Moreover, the author gives various examples of discrete groups which are not measure equivalent to the mapping class groups. In the course of the proof, the author investigates 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 and, using this investigation, proves that the mapping class group of a compact orientable surface is exact. Table of Contents  Introduction
 Property A for the curve complex
 Amenability for the actions of the mapping class group on the boundary of the curve complex
 Indecomposability of equivalence relations generated by the mapping class group
 Classification of the mapping class groups in terms of measure equivalence I
 Classification of the mapping class groups in terms of measure equivalence II
 Appendix A. Amenability of a group action
 Appendix B. Measurability of the map associating image measures
 Appendix C. Exactness of the mapping class group
 Appendix D. The cost and \(\ell^{2}\)Betti numbers of the mapping class group
 Appendix E. A grouptheoretic argument for Chapter 5
 Bibliography
 Index
