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 Izv. RAN. Ser. Mat., 2003, Volume 67, Issue 1, Pages 59–82 (Mi izv418)

Some new results on Borel irreducibility of equivalence relations

V. G. Kanovei, M. Reeken

Abstract: 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.

DOI: https://doi.org/10.4213/im418

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English version:
Izvestiya: Mathematics, 2003, 67:1, 55–76

Bibliographic databases:

UDC: 510.225
MSC: 03E15, 54E50

Citation: V. G. Kanovei, M. Reeken, “Some new results on Borel irreducibility of equivalence relations”, Izv. RAN. Ser. Mat., 67:1 (2003), 59–82; Izv. Math., 67:1 (2003), 55–76

Citation in format AMSBIB
\Bibitem{KanRee03} \by V.~G.~Kanovei, M.~Reeken \paper Some new results on Borel irreducibility of equivalence relations \jour Izv. RAN. Ser. Mat. \yr 2003 \vol 67 \issue 1 \pages 59--82 \mathnet{http://mi.mathnet.ru/izv418} \crossref{https://doi.org/10.4213/im418} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1957916} \zmath{https://zbmath.org/?q=an:1068.03036} \transl \jour Izv. Math. \yr 2003 \vol 67 \issue 1 \pages 55--76 \crossref{https://doi.org/10.1070/IM2003v067n01ABEH000418} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000185513200004} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33748479303} 

• http://mi.mathnet.ru/eng/izv418
• https://doi.org/10.4213/im418
• http://mi.mathnet.ru/eng/izv/v67/i1/p59

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Rosendal C., “Cofinal families of Borel equivalence relations and quasiorders”, J. Symbolic Logic, 70:4 (2005), 1325–1340
2. V. G. Kanovei, V. A. Lyubetskii, “A Cofinal Family of Equivalence Relations and Borel Ideals Generating Them”, Proc. Steklov Inst. Math., 252 (2006), 85–103
3. V. G. Kanovei, V. A. Lyubetskii, “Borel reducibility as an additive property of domains”, J. Math. Sci. (N. Y.), 158:5 (2009), 708–712
4. Thomas S., “On the complexity of the quasi-isometry and virtual isomorphism problems for finitely generated groups”, Groups Geom. Dyn., 2:2 (2008), 281–307
5. Thomas S., “A remark on the Higman–Neumann-Neumann embedding theorem”, J. Group Theory, 12:4 (2009), 561–565
6. Thomas S., “A Descriptive View of Combinatorial Group Theory”, Bull Symbolic Logic, 17:2 (2011), 252–264
7. Calderoni F., Thomas S., “The Bi-Embeddability Relation For Countable Abelian Groups”, Trans. Am. Math. Soc., 371:3 (2019), 2237–2254
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