The Descriptive Set Theory of Polish Group ActionsIn 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. |
Contents
0 DESCRIPTIVE SET THEORY | 1 |
1 POLISH GROUPS | 3 |
2 ACTIONS OF POLISH GROUPS | 13 |
3 EQUIVALENCE RELATIONS | 33 |
4 INVARIANT MEASURES AND PARADOXICAL DECOMPOSITIONS | 44 |
5 BETTER TOPOLOGIES | 53 |
Common terms and phrases
action of G algebraic assume axioms Borel action Borel equivalence relation Borel G-space Borel sets Borel structure Borel-measurable C-measurable canonical clearly closed subgroup complete metric continuous action continuous with respect Corollary countable basis countable models definable denote descriptive set theory embedding equivalence classes equivalence relation induced exists finer topology finite function G₁ group G homeomorphism implies invariant Borel set Kechris 95 L-structure language left action left-invariant Lemma Let G locally compact logic action Math measure metric space model theory modulo N(E N₁ nonempty open open nbhd orbit cardinal perfectly many orbits pointset Polish group actions Polish space Polish topology proof property of Baire proved relation symbol result Sami 94 second countable separable metric space separable metrizable sequence standard Borel space subgroup of G t-open Theorem topological group Topological Vaught Conjecture TVC2 Vaught Conjecture w₁