Recent developments in the theory of Borel reducibility

Greg Hjorth¹ ; Alexander S. Kechris²

¹ Department of Mathematics UCLA Los Angeles, CA 90095-1555, U.S.A.
² Department of Mathematics Caltech Pasadena, CA 91125, U.S.A.

Fundamenta Mathematicae, Tome 170 (2001) no. 1, pp. 21-52

Résumé

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.

Détail

DOI : 10.4064/fm170-1-2

Keywords: vitali equivalence relation product countably many copies dichotomy theorems borel equivalence relations proved first borel equivalence relation borel reducible either reducible else reducible second borel equivalence relation induced borel action closed subgroup infinite symmetric group admits invariant metric either reducible countable borel equivalence relation else reducible survey number results conjectures concerning global structure reducibility borel equivalence relations