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.
General
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