this condition, in certain situations, like, e.g., when E is given by an action
of the free group with infinitely many generators, the cohomology relation is
not smooth. Thus in the E0-ergodic case, there is an additional dichotomy
having to do with the structure of the target groups. This is the topic of
Section 29. Finally in Section 30 we deal with the special case of actions of
groups with property (T) and discuss some recent results of Popa on cocycle
superrigidity. We also establish in the above sections some new character-
izations of amenable and property (T) groups, that, in particular, extend
earlier results of Schmidt and also show that there is a positive link between
the E0-ergodicity of an equivalence relation E and the Polishness of its outer
automorphism group, Out(E), an issue raised by Jones-Schmidt. They have
pointed out that E0-ergodicity does not in general imply that Out(E) is
Polish but we show in Section 29 that one has a positive implication when
E is induced by a free action of a group with the minimal condition on
Appendices A I present background material concerning Hilbert spaces
and tensor products, Gaussian probability spaces and the Wiener chaos de-
composition, several relevant aspects of the theory of unitary representations
(including unitary representations of abelian groups and induced representa-
tions as well as some basic results about the space of unitary representations
of a group) and finally semidirect products of groups. We also include, in
Appendix E, a detailed proof of the standard result that any unitary repre-
sentation of a countable group is a subrepresentation of the Koopman rep-
resentation associated with some measure preserving action of that group.
(C) I would like to thank Mikl´ os Ab´ ert, Scot Adams, Oleg Ageev, Sergey
Bezuglyi, Andr´ es Caicedo, Clinton Conley, Damien Gaboriau, Sergey Gefter,
Eli Glasner, Greg Hjorth, Adrian Ioana, John Kittrell, Alain Louveau, Ben
Miller, Sorin Popa, Dinakar Ramakrishnan, Sergey Sinelshchikov, Simon
Thomas, Todor Tsankov and Vladimir Zhuravlev for their comments on
this book or useful conversations on matters related to it. Finally, I would
like to thank Leona Kershaw for typing the manuscript.
Research for this work was partially supported by NSF Grant DMS-
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