Introduction

(A) The original motivation for this work came from the study of cer-

tain 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 con-

cerning the structure of the outer automorphism group of a countable mea-

sure 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 T¨ ornquist on the existence of many non-orbit equivalent ergodic

actions of certain non-amenable groups, v) Popa’s recent work on cocycle

superrigidity.

Despite the apparent diversity of the subjects treated in these works,

we gradually realized that they can be understood within a rather unified

framework. This is the study of the global structure of the space A(Γ,X,µ)

of measure preserving actions of a countable group Γ on a standard measure

space (X, µ) and the canonical action of the automorphism group Aut(X, µ)

of (X, µ) by conjugation on A(Γ,X,µ) as well as the study of the global

structure of the space of cocycles and certain canonical actions on it. Our

goal here is to explore this point of view by presenting (a) earlier results,

sometimes in new formulations or with new proofs, (b) new theorems, and

finally (c) interesting open questions that are suggested by this approach.

(B) The book is divided into three chapters, the first consisting of Sec-

tions 1–9, the second of Sections 10–18, and the third of Sections 19–30.

There are also nine appendices.

In the first chapter, we study the automorphism group Aut(X, µ) of

a standard measure space (i.e., the group of measure preserving automor-

phisms of (X, µ)) and various subgroups associated with measure preserving

equivalence relations. Note that Aut(X, µ) can be also identified with the

space A(Z,X,µ) of measure preserving Z-actions. Sections 1, 2 review some

basic facts about the group Aut(X, µ). In Section 2 we also show that

the class of mild mixing transformations in Aut(X, µ) is co-analytic but

not Borel (a result also proved independently by Robert Kaufman). This

is in contrast with the well-known fact that the ergodic, weak mixing and

(strong) mixing transformations are, resp., Gδ,Gδ and Fσδ sets. In Section

ix