**Memoirs of the American Mathematical Society**

2008;
190 pp;
Softcover

MSC: Primary 20; 37;

Print ISBN: 978-0-8218-4196-9

Product Code: MEMO/196/916

List Price: $80.00

AMS Member Price: $48.00

MAA member Price: $72.00

**Electronic ISBN: 978-1-4704-0522-9
Product Code: MEMO/196/916.E**

List Price: $80.00

AMS Member Price: $48.00

MAA member Price: $72.00

# The Mapping Class Group from the Viewpoint of Measure Equivalence Theory

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*Yoshikata Kida*

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

# Table of Contents

## The Mapping Class Group from the Viewpoint of Measure Equivalence Theory

- Contents v6 free
- Chapter 1. Introduction 110 free
- Chapter 2. Property A for the curve complex 918 free
- Chapter 3. Amenability for the action of the mapping class group on the boundary of the curve complex 2736
- Chapter 4. Indecomposability of equivalence relations generated by the mapping class group 4756
- 1. Construction of Busemann functions and the MIN set map 4958
- 2. Preliminaries on discrete measured equivalence relations 6069
- 3. Reducible elements in the mapping class group 6574
- 4. Subrelations of the two types: irreducible and amenable ones and reducible ones 6776
- 5. Canonical reduction systems for reducible subrelations 7786
- 6. Indecomposability of equivalence relations generated by actions of the mapping class group 8291
- 7. Comparison with hyperbolic groups 92101

- Chapter 5. Classification of the mapping class groups in terms of measure equivalence I 95104
- Chapter 6. Classification of the mapping class groups in terms of measure equivalence II 123132
- Appendix A. Amenability of a group action 157166
- Appendix B. Measurability of the map associating image measures 167176
- Appendix C. Exactness of the mapping class group 169178
- Appendix D. The cost and l[sup(2)]-Betti numbers of the mapping class group 173182
- Appendix E. A group-theoretic argument for Chapter 5 179188
- Bibliography 183192
- Index 187196 free