This book belongs to the topic of Descriptive Set Theory.
Over the last 20 years, the theory of Borel equivalence relations and related topics have been very active areas of research in set theory and have important interactions with other fields of mathematics, like ergodic theory and topological dynamics, group theory, combinatorics, functional analysis, and model theory. The book presents, for the first time in mathematical literature, all major aspects of this theory and its applications.
This book may be of interest to a wide spectrum of mathematicians working in set theory as well as the other areas mentioned. It provides a systematic exposition of results that so far have been only available in journals or are even unpublished. The book presents unified and in some cases significantly streamlined proofs of several difficult results, especially dichotomy theorems. It has rather minimal overlap with other books published in this subject.
Author(s): Vladimir Kanovei
Series: University Lecture Series, Volume 44
Publisher: American Mathematical Society
Year: 2008
Language: English
Commentary: Scanned, DjVu'ed, OCR'ed, TOC by Envoy
Pages: 240
City: Providence, Rhode Island
Cover ......Page 1
Contents ......Page 4
Preface ......Page 8
Introduction ......Page 10
1.1. Polish spaces ......Page 16
1.2. Pointsets. Borel sets ......Page 17
1.4. Analytic formulas ......Page 19
1.5. Transformation of analytic formulas ......Page 21
1.6. Effective hierarchies of pointsets ......Page 22
1.7. Characterization of Sigma^0_1 sets ......Page 23
1.8. Classifying functions ......Page 24
1.9. Closure properties ......Page 25
2.1. Trees and ranks ......Page 28
2.2. Trees and sets of the first projective level ......Page 31
2.3. Reduction and separation ......Page 32
2.4. Uniformization and Kreisel Selection ......Page 33
2.5. Universal sets ......Page 36
2.6. Good universal sets ......Page 38
2.7. Reflection ......Page 39
2.8. Enumeration of Delta^1_1 sets ......Page 40
2.9. Coding Borel sets ......Page 42
2.10. Choquet property of Sigma^1_1 and the Gandy-Harrington topology ......Page 43
2.11. Sets with countable sections ......Page 45
2.12. Applications for Borel sets ......Page 47
3.2. Reducibility of ideals ......Page 50
3.3. P-ideals and submeasures ......Page 52
3.4. Polishable ideals ......Page 53
3.5. Characterization of polishable ideals ......Page 54
3.6. Summable and density ideals ......Page 56
3.8. Some other ideals ......Page 58
4.1. Some examples of Borel equivalence relations ......Page 60
4.2. Operations on equivalence relations ......Page 61
4.3. Orbit equivalence relations of group actions ......Page 63
4.4. Some examples of orbit equivalence relations ......Page 64
4.5. Probability measures ......Page 66
4.6. Invariant and ergodic measures ......Page 67
5.1. Borel reducibility ......Page 72
5.2. Injective Borel reducibility-embedding ......Page 73
5.3. Borel, continuous, and Baire measurable reductions ......Page 74
5.4. Additive reductions ......Page 75
5.5. Diagram of Borel reducibility of key equivalence relations ......Page 76
5.6. Reducibility and irreducibility on the diagram ......Page 77
5.7. Dichotomy theorems ......Page 79
5.8. Borel ideals in the structure of Borel reducibility ......Page 80
6.1. Equivalence relations E3 and T2 ......Page 82
6.2. Discretization and generation by ideals ......Page 83
6.3. Summables irreducible to density-0 ......Page 85
6.4. How to eliminate forcing ......Page 88
6.5. The family ell^p ......Page 89
6.6. ell^p: maximal K_sigma ......Page 91
7.1. Several types of equivalence relations ......Page 94
7.2. Smooth and below ......Page 95
7.3. Assembling countable equivalence relations ......Page 97
7.4. Countable equivalence relations and group actions ......Page 98
7.5. Non-hyperfinite countable equivalence relations ......Page 99
7.6. A sufficient condition of essential countability ......Page 102
8.1. Hyperfinite equivalence relations: The characterization theorem ......Page 104
8.2. Proof of the characterization theorem ......Page 105
8.3. Hyperfiniteness of tail equivalence relations ......Page 110
8.4. Classification modulo Borel isomorphism ......Page 112
8.5. Remarks on the classification theorem ......Page 113
8.6. Which groups induce hyperfinite equivalence relations? ......Page 115
Chapter 9. More on countable equivalence relations ......Page 116
9.1. Amenable groups ......Page 117
9.2. Amenable equivalence relations ......Page 118
9.3. Hyperfiniteness and amenability ......Page 120
9.4. TYeeable equivalence relations ......Page 121
9.5. Above treeable. Free Borel countable equivalence relations ......Page 122
10.1. The 1st dichotomy theorem ......Page 128
10.2. Splitting system ......Page 130
10.4. 2nd dichotomy theorem ......Page 131
10.5. Restricted product forcing ......Page 134
10.6. Splitting system ......Page 135
10.7. Construction of a splitting system ......Page 136
10.8. The ideal of E0-small sets ......Page 137
10.9. A forcing notion associated with E0 ......Page 139
11.1. Ideals below I1 ......Page 142
11.2. E1: hypersmoothness and non-countability ......Page 144
11.3. 3rd dichotomy ......Page 145
11.5. Case 2 ......Page 147
11.6. The construction ......Page 149
11.7. A forcing notion associated with E1 ......Page 151
11.8. Above E1 ......Page 152
12.1. Infinite symmetric group S_infty and isomorphisms ......Page 156
12.2. Borel invariant sets ......Page 157
12.3. Equivalence relations classifiable by countable structures ......Page 158
12.4. Reduction to countable graphs ......Page 159
12.5. Reduction of Borel classifiability to T_xi ......Page 160
13.1. Local orbits and turbulence ......Page 164
13.2. Shift actions of summable ideals are turbulent ......Page 165
13.3. Ergodicity ......Page 166
13.4. "Generic" reduction to T_xi ......Page 167
13.5. Ergodicity of turbulent actions w.r.t. T_xi ......Page 169
13.6. Inductive step of countable power ......Page 170
13.8. Other inductive steps ......Page 172
13.9. Applications to the shift action of ideals ......Page 173
14.1. Continual assembling of equivalence relations ......Page 176
14.2. The two cases ......Page 178
14.3. Case 1 ......Page 180
14.4. Case 2 ......Page 181
14.5. Splitting system ......Page 182
14.6. The embedding ......Page 183
14.8. The construction of a splitting system: the step ......Page 184
14.9. A forcing notion associated with E3 ......Page 187
15.1. Classification of summable ideals and equivalence relations ......Page 190
15.2. Grainy sets and the two cases ......Page 191
15.3. Case 1 ......Page 192
15.4. Case 2 ......Page 194
15.5. The construction of a splitting system ......Page 195
15.6. A forcing notion associated with E2 ......Page 196
16.1. c0-equalities: definition ......Page 200
16.2. Some examples and simple results ......Page 201
16.3. c0-equalities and additive reducibility ......Page 202
16.4. A largest c0-equality ......Page 203
16.5. Classification ......Page 204
16.6. LV-equalities ......Page 206
16.7. Non-sigma-compact case ......Page 209
17.1. The definition of pinned equivalence relations ......Page 212
17.3. Fubini product of pinned equivalence relations is pinned ......Page 214
17.4. Complete left-invariant actions induce pinned relations ......Page 215
17.5. All equivalence relations with Sigma^0_3 classes are pinned ......Page 216
17.6. Another family of pinned ideals ......Page 217
18.1. Trees ......Page 220
18.2. Louveau-Rosendal transform ......Page 221
18.3. Embedding and equivalence of normal trees ......Page 223
18.4. Reduction to Borel ideals: first approach ......Page 225
18.5. Reduction to Borel ideals: second approach ......Page 227
18.6. Some questions ......Page 230
A.1. Models of a fragment of ZFC ......Page 232
A.3. Forcing over countable models ......Page 234
A.4. Cohen forcing ......Page 236
A.5. Gandy-Harrington forcing ......Page 237
Bibliography ......Page 240
Index ......Page 244
Titles in this series ......Page 250
Back cover ......Page 252
