Recent developments in the theory of Borel reducibility

Abstract

Let E_0 be the Vitali equivalence relation and E_3 the product of countably many copies of E_0. Two new dichotomy theorems for Borel equivalence relations are proved. First, for any Borel equivalence relation E that is (Borel) reducible to E_3, either E is reducible to E_0 or else E_3 is reducible to E. Second, if E is a Borel equivalence relation induced by a Borel action of a closed subgroup of the infinite symmetric group that admits an invariant metric, then either E is reducible to a countable Borel equivalence relation or else E_3 is reducible to E. We also survey a number of results and conjectures concerning the global structure of reducibility on Borel equivalence relations.