x INTRODUCTION

3 we discuss the full group [E] ⊆ Aut(X, µ) of a measure preserving count-

able Borel equivalence relation E on (X, µ). In Section 4 we give a detailed

proof of Dye’s reconstruction theorem, which asserts that the equivalence

relation E is determined up to (measure preserving) isomorphism by [E]

as an abstract group. In Section 5 we give a new method for proving the

turbulence property of the conjugacy action of Aut(X, µ) on the set ERG

of ergodic transformations in Aut(X, µ), originally established by Foreman-

Weiss, and use this method to prove other turbulence results in the context

of full groups. We also extend the work of Hjorth and Foreman-Weiss on

non-classification by countable structures of weak mixing transformations in

Aut(X, µ), up to conjugacy or unitary (spectral) equivalence, to the case of

mixing transformations (and obtain more precise information in this case).

In Sections 6, 7 we review the basic properties of the automorphism group

N[E] of a measure preserving countable Borel equivalence relation E on

(X, µ) and its outer automorphism group, Out(E), and establish (in Section

7) the turbulence of the latter, when E is hyperfinite (and even in more

general situations). Understanding when Out(E) is a Polish group (with

respect to the canonical topology discussed in Section 7) is an interesting

open problem raised in work of Jones-Schmidt. In Section 8 we establish

a connection between the Polishness of Out(E) and Gaboriau’s theory of

costs and use it to obtain a partial answer to this problem. In Section 9 we

discuss Effros’ notion of inner amenability and its relationship with the open

problem of Schmidt of whether this property of a group is characterized by

the failure of Polishness of the outer automorphism group of some equiva-

lence relation induced by a free, measure preserving, ergodic action of the

group. Some known and new results related to this concept and Schmidt’s

problem are presented here.

In the second chapter, we start (in Section 10) with establishing some

basic properties of the space A(Γ,X,µ) of measure preserving actions of a

countable group Γ on (X, µ). For finitely generated groups, we also calcu-

late an upper bound for the descriptive complexity of the cost function on

this space and use this to show that the generic action realizes the cost of

the group. In Section 11 we recall several known ergodic theoretic char-

acterizations of groups with property (T) or the Haagerup Approximation

Property (HAP). In Section 12 we study the structure of the space of er-

godic actions (and some of its subspaces) in A(Γ,X,µ), recasting in this

context characterizations of Glasner-Weiss and Glasner concerning property

(T) or the HAP, originally formulated in terms of the structure of extreme

points in the space of invariant measures. In Section 13 we study, using

the method introduced in Section 5, turbulence of conjugacy in the space

of actions and also discuss work of Hjorth and Foreman-Weiss concerning

non-classification by countable structures of such actions, up to conjugacy.

We also prove an analogous result for unitary (spectral) equivalence. In Sec-

tion 14, we present the essence of Hjorth’s result on the existence of many