Countable Borel Equivalence Relations

This paper is a contribution to a new direction in descriptive set theory that is being extensively pursued over the last decade or so. It deals with 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 study 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. (For an extensive discussion of these matters, see, e.g., Hjorth [00], Kechris [99, 00a].) This theory is developed within the context of descriptive set theory, which provides the basic underlying concepts and methods. On the other hand, in view of its broad scope, there are natural interactions of it with other areas of mathematics, such as model theory, recursion theory, the theory of topological groups and their representations, topological dynamics, ergodic theory, and operator algebras. Classically, in various branches of dynamics one studies actions of the group of integers Z, reals R, Lie groups, etc. More generally, one can consider definable (e.g., continuous, Borel, etc.) actions of Polish (i.e., separable, completely metrizable topological) groups on standard Borel spaces (i.e., Polish spaces equipped with their σ-algebra of Borel sets). Most of our emphasis in this paper though will be on Borel actions of Polish locally compact groups. One of the main problems concerning a given definable action of a Polish group G on a standard Borel space X is the complete classification of members of X, up to orbit equivalence, by invariants. (Orbit equivalence being the equivalence relation induced by the orbits of the action.) This is a special case of the more general problem of completely classifying elements