The Descriptive Set Theory of Polish Group Actions

by Howard Becker , Alexander S. Kechris

• ISBN13: 9780521576055
• ISBN10: 0521576059
• Format: Paperback
• Copyright: 1996-12-28
• Publisher: Cambridge University Press

Summary

In this book the authors present their research into the foundations of the theory of Polish groups and the associated orbit equivalence relations. The particular case of locally compact groups has long been studied in many areas of mathematics. Non-locally compact Polish groups occur naturally as groups of symmetries in such areas as logic (especially model theory), ergodic theory, group representations, and operator algebras. Some of the topics covered here are: topological realizations of Borel measurable actions; universal actions; applications to invariant measures; actions of the infinite symmetric group in connection with model theory (logic actions); dichotomies for orbit spaces (including Silver, Glimm-Effros type dichotomies and the topological Vaught conjecture); descriptive complexity of orbit equivalence relations; definable cardinality of orbit spaces.

Table of Contents

PREFACE	vii	(1)
INTRODUCTION	viii
0. DESCRIPTIVE SET THEORY	1	(2)
1. POLISH GROUPS	3	(10)
1.1 Metrizable groups	3	(1)
1.2 Polish groups	3	(4)
1.3 Examples	7	(1)
1.4 Universal Polish groups	7	(1)
1.5 Some facts about the symmetric group S(XXX)	8	(3)
1.6 Standard Borel and Polishable groups	11	(2)
2. ACTIONS OF POLISH GROUPS	13	(20)
2.1 G-spaces	13	(1)
2.2 Polish G-spaces	14	(3)
2.3 Borel G-spaces	17	(3)
2.4 Examples	20	(2)
2.5 The logic actions	22	(1)
2.6 Universal actions	22	(6)
2.7 Universal actions for closed subgroups of S(XXX)	28	(5)
3. EQUIVALENCE RELATIONS	33	(11)
3.1 Generalities	33	(1)
3.2 Equivalence relations induced by actions	34	(1)
3.3 The Silver Dichotomy	34	(1)
3.4 The Glimm-Effros Dichotomy	35	(6)
3.5 Universal equivalence relations	41	(3)
4. INVARIANT MEASURES AND PARADOXICAL DECOMPOSITIONS	44	(9)
4.1 Tarski's Theorem	44	(1)
4.2 Countable decompositions	44	(1)
4.3 Nadkarni's Theorem	45	(1)
4.4 Proof of 4.2.1	46	(1)
4.5 Sketch of proof of Nadkarni's Theorem	47	(5)
4.6 Concluding remarks and problems	52	(1)
5. BETTER TOPOLOGIES	53	(29)
5.1 Finer topologies and Borel sets	53	(5)
5.2 Topological realization of Borel G-spaces	58	(9)
5.3 Topological realization of definable G-spaces	67	(4)
5.4 Finer topologies on G-spaces	71	(11)
6. MODEL THEORY AND THE VAUGHT CONJECTURE	82	(16)
6.1 Background on the Vaught Conjecture	82	(5)
6.2 The Topological Vaught Conjecture	87	(9)
6.3 Atomic models	96	(2)
7. ACTIONS WITH BOREL ORBIT EQUIVALENCE RELATIONS	98	(18)
7.1 Characterizations	98	(5)
7.2 Some effective considerations	103	(1)
7.3 Decompositions	104	(4)
7.4 Tame groups	108	(2)
7.5 Normalizers	110	(6)
8. DEFINABLE CARDINALITY	116	(6)
8.1 Orbit cardinality	116	(3)
8.2 Orbit cardinality for specific groups	119	(3)
References	122	(10)
Index	132