Chapter 1. Introduction ......................................... 1
Chapter 2. Property A for the curve complex ..................... 9
1. Geometry of the curve complex ............................. 9
2. Generalities for property A .............................. 12
3. Property A for the curve complex ......................... 13
4. Exceptional surfaces ..................................... 20
Chapter 3. Amenability for the action of the mapping class
group on the boundary of the curve complex .......... 27
1. The mapping class group and the Thurston boundary ........ 27
2. The boundary at infinity of the curve complex ............ 32
3. Amenability for the actions of the mapping class group ... 35
4. The boundary of the curve complex for an exceptional
surface .................................................. 42
Chapter 4. Indecomposability of equivalence relations
generated by the mapping class group ................ 47
1. Construction of Busemann functions and the MIN set map ... 49
2. Preliminaries on discrete measured equivalence
relations ................................................ 60
3. Reducible elements in the mapping class group ............ 65
4. Subrelations of the two types: irreducible and
amenable ones and reducible ones ......................... 67
5. Canonical reduction systems for reducible subrelations ... 77
6. Indecomposability of equivalence relations generated by
actions of the mapping class group ....................... 82
7. Comparison with hyperbolic groups ........................ 92
Chapter 5. Classification of the mapping class groups in
terms of measure equivalence I ...................... 95
1. Reducible subrelations, revisited ........................ 97
2. Irreducible and amenable subsurfaces .................... 105
3. Amenable, reducible subrelations ........................ 107
4. Classification .......................................... 110
Chapter 6. Classification of the mapping class groups in
terms of measure equivalence II .................... 123
1. Geometric lemmas ........................................ 125
2. Families of subrelations satisfying the maximal
condition ............................................... 127
3. Application I (Invariance of complexity under measure
equivalence) ............................................ 133
4. Application II (The case where complexity is odd) ....... 136
5. Application III (The case where complexity is even) ..... 146
Appendix A. Amenability of a group action ..................... 157
1. Notation ................................................ 157
2. Existence of invariant means ............................ 159
3. The fixed point property ................................ 161
Appendix B. Measurabihty of the map associating image
measures .......................................... 167
Appendix C. Exactness of the mapping class group .............. 169
Appendix D. The cost and ℓ2-Betti numbers of the mapping
class group ....................................... 173
1. The cost of the mapping class group ..................... 173
2. The ℓ2-Betti numbers of the mapping class group ......... 176
Appendix E. A group-theoretic argument for Chapter 5 .......... 179
Bibliography .................................................. 183
Index ......................................................... 187
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