Geodesic
Some new results on Borel irreducibility of equivalence relations
Izvestiya. Mathematics , Tome 67 (2003) no. 1, pp. 55-76.

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We prove that orbit equivalence relations (ERs, for brevity) of generically turbulent Polish actions are not Borel reducible to ERs of a family which includes Polish actions of $S_\infty$ (the group of all permutations of $\mathbb N$ and is closed under the Fubini product modulo the ideal Fin of all finite sets and under some other operations. We show that $\mathsf T_2$ (an equivalence relation called the equality of countable sets of reals is not Borel reducible to another family of ERs which includes continuous actions of Polish CLI groups, Borel equivalence relations with $\mathbf G_{\delta\sigma}$ classes and some ideals, and is closed under the Fubini product modulo Fin. These results and their corollaries extend some earlier irreducibility theorems of Hjorth and Kechris. 
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V. G. Kanovei; M. Reeken. Some new results on Borel irreducibility of equivalence relations. Izvestiya. Mathematics , Tome 67 (2003) no. 1, pp. 55-76. http://geodesic.mathdoc.fr/item/IM2_2003_67_1_a3/

[1] Becker H., Kechris A. S., The descriptive set theory of Polish group actions, Cambridge University Press, Cambridge, 1996 | MR | Zbl

[2] Farah I., “Analytic quotients: theory of liftings for quotients over analytic ideals on the integers”, Mem. Amer. Math. Soc., 148(702) (2000) ; XVI, 177 pp. | MR

[3] Friedman H., Stanley L., “A Borel reducibility theory for classes of countable structures”, J. Symbolic Logic, 54:3 (1989), 894–914 | DOI | MR | Zbl

[4] Friedman H., “Borel and Baire reducibility”, Fund. Math., 164:1 (2000), 61–69 | MR | Zbl

[5] Hjorth G., “Orbit cardinals: on the effective cardinalities arising as quotient spaces of the form $X/G$ where $G$ acts on a Polish space $X$”, Israel J. Math., 111 (1999), 221–261 | DOI | MR | Zbl

[6] Hjorth G., Classification and orbit equivalence relations, American Math. Society, Providence, 2000 | MR | Zbl

[7] Hjorth G., Kechris A. S., “New dichotomies for Borel equivalence relations”, Bull. Symbolic Logic, 3:3 (1997), 329–346 | DOI | MR | Zbl

[8] Jackson S., Kechris A. S., Louveau A., “Countable Borel equivalence relations”, J. Math. Logic, 2:1 (2002), 1–80 | DOI | MR | Zbl

[9] Kanovei V., “An Ulm-type classification theorem for equivalence relations in Solovay model”, J. Symbolic Logic, 62:4 (1997), 1333–1351 | DOI | MR | Zbl

[10] Kanovei V., “Ulm classification of analytic equivalence relations in generic universes”, Math. Logic Quart., 44:3 (1998), 287–303 | DOI | MR | Zbl

[11] Kechris A. S., “Rigidity properties of Borel ideals on the integers”, Topology Appl., 85:1–3 (1998), 195–205 | DOI | MR | Zbl

[12] Kechris A. S., “New directions in descriptive set theory”, Bull. Symbolic Logic, 5:2 (1999), 161–174 | DOI | MR | Zbl

[13] Kechris A. S., “Actions of Polish groups and classification problems”, Analysis and Logic, London Mathematical Society Lecture Note Series, Cambridge University Press, 2002, 168–237 | MR

[14] Louveau A., “A separation theorem for $\Sigma_1^1$ sets”, Trans. Amer. Math. Soc., 260 (1980), 363–378 | DOI | MR

[15] Solecki S., “Analytic ideals and their applications”, Ann. Pure Appl. Logic, 99:3 (1999), 51–72 | DOI | MR | Zbl