The classification of hypersmooth Borel equivalence relations

Abstract

This paper is a contribution to the study of Borel equivalence relations in standard Borel spaces, i.e., Polish spaces equipped with their Borel structure. A class of such equivalence relations which has received particular attention is the class of hyperfinite Borel equivalence relations. These can be defined as the increasing unions of sequences of Borel equivalence relations all of whose equivalence classes are finite or, as it turns out, equivalently those induced by the orbits of a single Borel auto-morphism. Hyperfinite equivalence relations have been classified in [DJK], under two notions of equivalence, Borel bi-reducibility, and Borel isomorphism.