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

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*Yoshikata Kida*

#### 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