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Global Aspects of Ergodic Group Actions
Alexander S. Kechris, California Institute of Technology, Pasadena, CA
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Mathematical Surveys and Monographs
2010; 237 pp; hardcover
Volume: 160
ISBN-10: 0-8218-4894-1
ISBN-13: 978-0-8218-4894-4
List Price: US$77
Member Price: US$61.60
Order Code: SURV/160
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The subject of this book is the study of ergodic, measure preserving actions of countable discrete groups on standard probability spaces. It explores a direction that emphasizes a global point of view, concentrating on the structure of the space of measure preserving actions of a given group and its associated cocycle spaces. These are equipped with canonical topological actions that give rise to the usual concepts of conjugacy of actions and cohomology of cocycles. Structural properties of discrete groups such as amenability, Kazhdan's property (T) and the Haagerup Approximation Property play a significant role in this theory as they have important connections to the global structure of these spaces. One of the main topics discussed in this book is the analysis of the complexity of the classification problems of conjugacy and orbit equivalence of actions, as well as of cohomology of cocycles. This involves ideas from topological dynamics, descriptive set theory, harmonic analysis, and the theory of unitary group representations. Also included is a study of properties of the automorphism group of a standard probability space and some of its important subgroups, such as the full and automorphism groups of measure preserving equivalence relations and connections with the theory of costs.

The book contains nine appendices that present necessary background material in functional analysis, measure theory, and group representations, thus making the book accessible to a wider audience.

Readership

Graduate students and research mathematicians interested in ergodic theory and descriptive set theory.

Reviews

"This monograph is written with a great exposition as well as with precise proofs and references. We would like to emphasize that the author has undertaken the tremendous job of collecting and systematizing the material from a huge number of different sources. The choice of material (both classical and new) is excellent. Furthermore, this seems to be the only book that discusses in such detail the topological structure of group actions and their subsets. This monograph could undoubtedly serve as a starting point for any researcher in the field."

-- Mathematical Reviews

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