Topics in Orbit Equivalence Book

I	Orbit equivalence	1
1	Group actions and equivalence relations	1
2	Invariant measures	3
3	Ergodicity	4
4	Isomorphism and orbit equivalence	6
II	Amenability and hyperfiniteness	7
5	Amenable groups	7
6	Hyperfiniteness	17
7	Dye's theorem	23
8	Quasi-invariant measures	33
9	Amenable equivalence relations	36
10	Amenability vs. hyperfiniteness	43
11	Groups of polynomial growth	47
12	Generic hyperfiniteness	50
13	Generic compressibility	51
III	Costs of equivalence relations and groups	55
14	Preliminaries	55
15	Countable Borel equivalence relations	56
16	More on invariant measures	57
17	Graphings of equivalence relations	58
18	Cost of an equivalence relation	59
19	Treeings of an equivalence relation	65
20	The cost of a smooth equivalence relation	66
21	The cost of a complete section	68
22	Cost and hyperfiniteness	69
23	Joins	73
24	Commuting equivalence relations	76
25	Subequivalence relations of finite index	81
26	Cheap equivalence relations	84
27	Free and amalgamated joins	85
28	Costs and free actions	98
29	Cost of a group	107
30	Treeable groups	113
31	Cost and amenable groups	115
32	Generating subgroups	117
33	Products	117
34	Subgroups of finite index	119
35	Cheap groups	120
36	Free and amalgamated products	123
37	HNN-extensions	125
38	A list of open problems	126