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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.
"I wholeheartedly recommend this book to anyone with an interest in group action and/or descriptive set theory." Klaas Pieter Hart, Mathematical Reviews
"This is an excellent book for anyone interested in Borel sets and analytic sets...in separable spaces whose topologies can be given by complete metrics...." Arlan Ramsay, Journal of Symbolic Logic
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