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The Mapping Class Group from the Viewpoint of Measure Equivalence Theory
Yoshikata Kida, Kyoto University, Japan, and Tohoku University, Sendai, Japan
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Memoirs of the American Mathematical Society
2008; 190 pp; softcover
Volume: 196
ISBN-10: 0-8218-4196-3
ISBN-13: 978-0-8218-4196-9
List Price: US$76
Individual Members: US$45.60
Institutional Members: US$60.80
Order Code: MEMO/196/916
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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 group-theoretic argument for Chapter 5
  • Bibliography
  • Index
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