Recent developments in the theory of Borel reducibility
Volume 170 / 2001
Fundamenta Mathematicae 170 (2001), 21-52 MSC: Primary 03E15. DOI: 10.4064/fm170-1-2
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.