non-orbit equivalent actions of property (T) groups as a basic property con-
cerning the topological structure of the conjugacy classes of ergodic actions
of such groups. We use Hjorth’s method to show that for such groups the set
of ergodic actions is clopen in the uniform topology and so is each conjugacy
class of ergodic actions. In Section 15 we study connectedness properties in
the space of actions, using again the method of Section 5. This illustrates
the close connection between local connectedness properties and turbulence.
We show, in particular, that the space A(Γ,X,µ) is path-connected in the
weak topology. We also contrast this to the work in Section 14 to point out
the interesting phenomenon that connectedness properties in the space of
actions of a group seem to be related to properties of the group, such as
amenability or property (T). In particular, for groups Γ with property (T),
we determine completely the path components of the space A(Γ,X,µ) in the
uniform topology. In Section 16 we discuss results of Popa concerning the
action of SL2(Z) on
These are used in Section 17, along with other ideas,
in the proof of a non-classification result of T¨ ornquist for orbit equivalence
of actions of non-abelian free groups. We also briefly discuss very recent re-
sults of Ioana, Epstein and Epstein-Ioana-Kechris-Tsankov that extend this
to arbitrary non-amenable groups. Finally, Section 18 contains a survey of
classification problems concerning group actions.
In the third chapter, we give in Section 19 a short introduction to the
properties of the group of group-valued random variables and then in Sec-
tions 20, 21 we discuss the space of cocycles of a group action or an equiv-
alence relation and some of the invariants associated with such cocycles,
like the associated Mackey action and the essential range. We also discuss
cocycles arising from reductions and homomorphisms of equivalence rela-
tions. Our primary interest is in cocycles with countable (discrete) targets.
The next two Sections 22 and 23 contain background material concerning
continuous and isometric group actions and Effros’ Theorem. The topol-
ogy of the space of cocycles is discussed in Section 24 and the study of the
global properties of the cohomology equivalence relation is the subject of
the final Sections 25–30. Section 25 contains some general properties of the
cohomology relation, and Section 26 is concerned with the hyperfinite case.
There is a large literature here but it is not our main focus in this work.
There is a fundamental dichotomy in the structure of the cohomology re-
lation for the cocycles of a given equivalence relation (or action), which in
some form is already present in the work of Schmidt for the case of cocycles
with abelian targets. In a precise sense, that is explained in these sections,
when an equivalence relation E is non E0-ergodic (or not strongly ergodic),
the structure of the cohomology relation on its cocycles is very complicated.
This is the subject in Section 27. On the other hand, when the equivalence
relation E is E0-ergodic (or strongly ergodic), then the cohomology relation
is simple, i.e., smooth, provided the target groups satisfy the so-called min-
imal condition on centralizers, discussed in Section 28 (and these include,
e.g., the abelian and the linear groups). If the target groups do not satisfy