The original motivation for this work came from the study of certain results in ergodic theory, primarily, but not exclusively, obtained over the last several years. These included: i) Work of Hjorth and Foreman-Weiss concerning the complexity of the problem of classification of ergodic measure preserving transformations up to conjugacy, ii) Various results concerning the structure of the outer automorphism group of a countable measure preserving equivalence relation, including, e.g., work of Jones-Schmidt, iii) Ergodic theoretic characterizations of groups with property (T) or the Haagerup Approximation Property, in particular results of Schmidt, Connes-Weiss, Jolissaint, and Glasner-Weiss, iv) Results of Hjorth, Popa, Gaboriau-Popa and Tornquist on the existence of many non-orbit equivalent ergodic actions of certain non-amenable groups, v) Popa’s recent work on cocycle superrigidity.

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