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 automor-
phism 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.
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