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.
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
Affiliations des auteurs :
Greg Hjorth
1
;
Alexander S. Kechris
2
1
Department of Mathematics UCLA Los Angeles, CA 90095-1555, U.S.A.
2
Department of Mathematics Caltech Pasadena, CA 91125, U.S.A.
Greg Hjorth; Alexander S. Kechris. Recent developments in the theory of Borel reducibility. Fundamenta Mathematicae, Tome 170 (2001) no. 1, pp. 21-52. doi: 10.4064/fm170-1-2
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