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
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