Some applications of the Adams-Kechris technique (Q2759039)

This paper is a contribution to the ongoing study of definable equivalence relations on Polish spaces. In particular the paper analyzes the technique used by \textit{S. Adams} and \textit{A. S. Kechris} in the proof that the partial order of Borel sets under inclusion can be embedded in the partial order of countable Borel equivalence relations under Borel reducibility [J. Am. Math. Soc. 13, 909-943 (2000; Zbl 0952.03057)]. This technique is isolated in section 2 of the present paper and yields several generalizations of the results contained in the paper by Adams and Kechris. NEWLINENEWLINENEWLINEAmong these generalizations are the fact that every analytic equivalence relation is Borel reducible to the equivalence relation of Borel bireducibility between countable Borel equivalence relations and that the same holds for any analytic or coanalytic equivalence relation with respect to the equivalence relation of \(\sigma (\Sigma^1_1)\) bireducibility between countable Borel equivalence relations. Other generalizations concern the possibility of applying the Adams-Kechris technique to uncountable Borel equivalence relation and in particular to those that are essentially so (that is, are not reducible to a countable Borel equivalence relation): the main result in this direction is the generalization of the Adams-Kechris results to turbulent equivalence relations. NEWLINENEWLINENEWLINEThe last section of the paper applies the Adams-Kechris technique to Borel isomorphisms of Borel automorphisms and mentions the major open problems in the area.