Contents
Introduction ix
Chapter I. Measure preserving automorphisms 1
1. The group Aut(X, µ) 1
2. Some basic facts about Aut(X, µ) 5
3. Full groups of equivalence relations 13
4. The Reconstruction Theorem 20
5. Turbulence of conjugacy 30
6. Automorphism groups of equivalence relations 40
7. The outer automorphism group 44
8. Costs and the outer automorphism group 48
9. Inner amenability 52
Chapter II. The space of actions 61
10. Basic properties 61
11. Characterizations of groups with property (T) and HAP 79
12. The structure of the set of ergodic actions 84
13. Turbulence of conjugacy in the ergodic actions 90
14. Conjugacy in ergodic actions of property (T) groups 98
15. Connectedness in the space of actions 105
16. The action of SL2(Z) on
T2
113
17. Non-orbit equivalent actions of free groups 117
18. Classifying group actions: A survey 122
Chapter III. Cocycles and cohomology 125
19. Group-valued random variables 125
20. Cocycles 129
21. The Mackey action and reduction cocycles 135
22. Homogeneous spaces and Effros’ Theorem 143
23. Isometric actions 145
24. Topology on the space of cocycles 148
25. Cohomology I: Some general facts 150
26. Cohomology II: The hyperfinite case 153
27. Cohomology III: The non E0-ergodic case 158
28. The minimal condition on centralizers 168
29. Cohomology IV: The E0-ergodic case 170
30. Cohomology V: Actions of property (T) groups 179
vii
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