We will discuss in this paper some aspects of a general program whose goal is the development of the theory of definable actions of Polish groups, the structure and classification of their orbit spaces, and the closely related study of definable equivalence relations. This work is motivated by basic foundational questions, like understanding the nature of complete classification of mathematical objects up to some notion of equivalence by invariants, and creating a mathematical framework for measuring the complexity of such classification problems. This theory, which has been growing rapidly over the last few years, is developed within the context of descriptive set theory, which provides the basic underlying concepts and methods. On the other hand, in view of the broad scope of this theory, there are natural interactions of it with other areas of mathematics, such as the theory of topological groups, topological dynamics, ergodic theory and its relationships with the theory of operator algebras, model theory, and recursion theory.

Classically, in various branches of dynamics one studies actions of the groups of integers ℤ, reals ℝ, Lie groups, or even more generally (second countable) locally compact groups. One of the goals of the theory is to expand this scope by considering the more comprehensive class of Polish groups (separable completely metrizable topological groups), which seems to be the widest class of well-behaved (for our purposes) groups and which includes practically every type of topological group we are interested in.

Let G be a countable group and X a Borel G-space. Then it is clear that is a Borel equivalence relation and every one of its equivalence class is countable.

Definition 4.1. A Borel equivalence relation E is countable if every equivalence class is countable.

We now have

Theorem 4.2 (Feldman-Moore [77]). The following are equivalent for each Borel equivalence relation E on a Polish space X:

1. (i) E is countable.

2. (ii) E = for some countable group G and a Borel G-space X.

Countable Borel equivalence relations have long been studied in ergodic theory and its relationship to the theory of operator algebras. One important observation is that many concepts of ergodic theory such as invariance, quasi-invariance (null set preservation), and ergodicity of measures depend only on the orbit equivalence relation and not the action inducing it. Also a countable Borel equivalence relation with an associated quasi-invariant probability measure gives rise to a canonical von Neumann algebra and this has important implications to classification problems of von Neumann algebras. See, for example, the survey Schmidt [90].

The simplest examples of countable Borel equivalence relations are those induced by Borel actions of the group of integers ℤ, i.e., by the orbits of a single Borel automorphism. These are also called hyperfinite in view of the following result.

Theorem 4.3 (Slaman-Steel [88], Weiss [84]). For each countable Borel equivalence relation E on a Polish space X the following are equivalent:

1. (i) E =, for some Borel ℤ-space X.

2. […]