Some new results on Borel irreducibility of equivalence relations

Vladimir Kanovei, Michael Reeken

Published 2002-03-11Version 1

We prove that orbit equivalence relations (ERs, for brevity) of generically turbulent Polish actions are not Borel reducible to ER s of a family which includes Polish actions of S_\infty, the group of all permutations of N, and is closed under the Fubini product modulo the ideal Fin of all finite sets, and some other operations. Our second main result shows that T_2, the equivalence relation called ``the equality of countable sets of the reals'', is not Borel reducible to another family of ERs which includes continuous actions of Polish CLI groups, Borel equivalence relations with G_\delta classes, some ideals, and is closed under the Fubini product over $Fin.$ Both results and their corollaries extend some earlier irreducibility theorems by Hjorth and Kechris.