Relevant bibliographies by topics / Borel complexity of equivalence relations

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Borel complexity of equivalence relations'

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Published: 13 April 2024

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Journal articles on the topic "Borel complexity of equivalence relations"

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Gao, Su, and Michael Ray Oliver. "Borel complexity of isomorphism between quotient Boolean algebras." Journal of Symbolic Logic 73, no. 4 (December 2008): 1328–40. http://dx.doi.org/10.2178/jsl/1230396922.

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In response to a question of Farah, “How many Boolean algebras are there?” [Far04], one of us (Oliver) proved that there are continuum-many nonisomorphic Boolean algebras of the form with I a Borel ideal on the natural numbers, and in fact that this result could be improved simultaneously in two directions:(i) “Borel ideal” may be improved to “analytic P-ideal”(ii) “continuum-many” may be improved to “E0-many”; that is, E0 is Borel reducible to the isomorphism relation on quotients by analytic P-ideals.See [Oli04].In [AdKechOO], Adams and Kechris showed that the relation of equality on Borel sets (and therefore, any Borel equivalence relation whatsoever) is Borel reducible to the equivalence relation of Borel bireducibility. (In somewhat finer terms, they showed that the partial order of inclusion on Borel sets is Borel reducible to the quasi-order of Borel reducibility.) Their technique was to find a collection of, in some sense, strongly mutually ergodic equivalence relations, indexed by reals, and then assign to each Borel set B a sort of “direct sum” of the equivalence relations corresponding to the reals in B. Then if B1, ⊆ B2 it was easy to see that the equivalence relation thus induced by B1 was Borel reducible to the one induced by B2, whereas in the opposite case, taking x to be some element of B / B2, it was possible to show that the equivalence relation corresponding to x, which was part of the equivalence relation induced by B1, was not Borel reducible to the equivalence relation corresponding to B2.
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MARKS, ANDREW. "The universality of polynomial time Turing equivalence." Mathematical Structures in Computer Science 28, no. 3 (July 13, 2016): 448–56. http://dx.doi.org/10.1017/s0960129516000232.

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We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin's ultrafilter on the Turing degrees.
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Ding, Longyun, and Su Gao. "Diagonal actions and Borel equivalence relations." Journal of Symbolic Logic 71, no. 4 (December 2006): 1081–96. http://dx.doi.org/10.2178/jsl/1164060445.

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AbstractWe investigate diagonal actions of Polish groups and the related intersection operator on closed subgroups of the acting group. The Borelness of the diagonal orbit equivalence relation is characterized and is shown to be connected with the Borelness of the intersection operator. We also consider relatively tame Polish groups and give a characterization of them in the class of countable products of countable abelian groups. Finally an example of a logic action is considered and its complexity in the Borel reducbility hierarchy determined.
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KRUPIŃSKI, KRZYSZTOF, ANAND PILLAY, and SŁAWOMIR SOLECKI. "BOREL EQUIVALENCE RELATIONS AND LASCAR STRONG TYPES." Journal of Mathematical Logic 13, no. 02 (October 31, 2013): 1350008. http://dx.doi.org/10.1142/s0219061313500086.

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The "space" of Lascar strong types, on some sort and relative to a given complete theory T, is in general not a compact Hausdorff topological space. We have at least three (modest) aims in this paper. The first is to show that spaces of Lascar strong types, as well as other related spaces and objects such as the Lascar group Gal L(T) of T, have well-defined Borel cardinalities (in the sense of the theory of complexity of Borel equivalence relations). The second is to compute the Borel cardinalities of the known examples as well as of some new examples that we give. The third is to explore notions of definable map, embedding, and isomorphism, between these and related quotient objects. We also make some conjectures, the main one being roughly "smooth if and only if trivial". The possibility of a descriptive set-theoretic account of the complexity of spaces of Lascar strong types was touched on in the paper [E. Casanovas, D. Lascar, A. Pillay and M. Ziegler, Galois groups of first order theories, J. Math. Logic1 (2001) 305–319], where the first example of a "non-G-compact theory" was given. The motivation for writing this paper is partly the discovery of new examples via definable groups, in [A. Conversano and A. Pillay, Connected components of definable groups and o-minimality I, Adv. Math.231 (2012) 605–623; Connected components of definable groups and o-minimality II, to appear in Ann. Pure Appl. Logic] and the generalizations in [J. Gismatullin and K. Krupiński, On model-theoretic connected components in some group extensions, preprint (2012), arXiv:1201.5221v1].
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KECHRIS, ALEXANDER S., ANDRÉ NIES, and KATRIN TENT. "THE COMPLEXITY OF TOPOLOGICAL GROUP ISOMORPHISM." Journal of Symbolic Logic 83, no. 3 (September 2018): 1190–203. http://dx.doi.org/10.1017/jsl.2018.25.

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AbstractWe study the complexity of the topological isomorphism relation for various classes of closed subgroups of the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Borel spaces. For profinite, locally compact, and Roelcke precompact groups, we show that the complexity is the same as the one of countable graph isomorphism. For oligomorphic groups, we merely establish this as an upper bound.
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Lecomte, Dominique. "On the complexity of Borel equivalence relations with some countability property." Transactions of the American Mathematical Society 373, no. 3 (December 10, 2019): 1845–83. http://dx.doi.org/10.1090/tran/7942.

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Calderoni, Filippo, Heike Mildenberger, and Luca Motto Ros. "Uncountable structures are not classifiable up to bi-embeddability." Journal of Mathematical Logic 20, no. 01 (September 6, 2019): 2050001. http://dx.doi.org/10.1142/s0219061320500014.

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Answering some of the main questions from [L. Motto Ros, The descriptive set-theoretical complexity of the embeddability relation on models of large size, Ann. Pure Appl. Logic 164(12) (2013) 1454–1492], we show that whenever [Formula: see text] is a cardinal satisfying [Formula: see text], then the embeddability relation between [Formula: see text]-sized structures is strongly invariantly universal, and hence complete for ([Formula: see text]-)analytic quasi-orders. We also prove that in the above result we can further restrict our attention to various natural classes of structures, including (generalized) trees, graphs, or groups. This fully generalizes to the uncountable case the main results of [A. Louveau and C. Rosendal, Complete analytic equivalence relations, Trans. Amer. Math. Soc. 357(12) (2005) 4839–4866; S.-D. Friedman and L. Motto Ros, Analytic equivalence relations and bi-embeddability, J. Symbolic Logic 76(1) (2011) 243–266; J. Williams, Universal countable Borel quasi-orders, J. Symbolic Logic 79(3) (2014) 928–954; F. Calderoni and L. Motto Ros, Universality of group embeddability, Proc. Amer. Math. Soc. 146 (2018) 1765–1780].
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HJORTH, GREG. "TREEABLE EQUIVALENCE RELATIONS." Journal of Mathematical Logic 12, no. 01 (June 2012): 1250003. http://dx.doi.org/10.1142/s0219061312500031.

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There are continuum many ≤B-incomparable equivalence relations induced by a free, Borel action of a countable non-abelian free group — and hence, there are 2α0 many treeable countable Borel equivalence relations which are incomparable in the ordering of Borel reducibility.
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JACKSON, S., A. S. KECHRIS, and A. LOUVEAU. "COUNTABLE BOREL EQUIVALENCE RELATIONS." Journal of Mathematical Logic 02, no. 01 (May 2002): 1–80. http://dx.doi.org/10.1142/s0219061302000138.

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This paper develops the foundations of the descriptive set theory of countable Borel equivalence relations on Polish spaces with particular emphasis on the study of hyperfinite, amenable, treeable and universal equivalence relations.
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Rosendal, Christian. "Cofinal families of Borel equivalence relations and quasiorders." Journal of Symbolic Logic 70, no. 4 (December 2005): 1325–40. http://dx.doi.org/10.2178/jsl/1129642127.

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AbstractFamilies of Borel equivalence relations and quasiorders that are cofinal with respect to the Borel reducibility ordering. ≤B, are constructed. There is an analytic ideal on ω generating a complete analytic equivalence relation and any Borel equivalence relation reduces to one generated by a Borel ideal. Several Borel equivalence relations, among them Lipschitz isomorphism of compact metric spaces, are shown to be Kσ complete.
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Dissertations / Theses on the topic "Borel complexity of equivalence relations"

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Robert, Simon. "Une approche par les groupes amples pour l’équivalence orbitale des actions minimales de Z sur l’espace de Cantor." Electronic Thesis or Diss., Lyon 1, 2023. http://www.theses.fr/2023LYO10142.

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Cette thèse s'inscrit dans le cadre de la dynamique topologique, branche des systèmes dynamiques s'intéressant aux comportements qualitatifs asymptotiques de transformations continues provenant d'une action de groupe ou de semigroupe sur un espace métrique usuellement compact. Par exemple, une question classique pourrait être de savoir si tel système dynamique admet des points récurrents, c'est à dire des points qui vont revenir arbitrairement proche de leur point de départ infiniment souvent sous la dynamique. Souvent, de par leur caractère qualitatif et asymptotique, ces propriétés ne dépendent pas précisément du système mais plutôt des orbites des points, i.e des positions qu'il vont atteindre. D'où la notion d'équivalence orbitale au coeur de cette thèse, qui consiste à considérer que, après identification des espaces sous-jacents, deux systèmes dont tous les points auraient les mêmes orbites seraient "qualitativement les mêmes". Au cours des années 90, Giordano Putnam et Skau ont réussi à établir grâce à des outils d'algèbre homologique une classification à équivalence orbitale près des systèmes dynamiques minimaux provenant d'actions de \Z sur l'espace de Cantor en termes à la fois de groupes pleins et de mesures invariantes. Ce résultat montre en particulier qu'il existe une infinité non-dénombrable de tels systèmes différents à équivalence orbitale près, ce qui contraste assez fortement avec le cadre de la théorie ergodique, domaine très proche s'intéressant aux systèmes dynamiques mesurés, dans lequel la combinaison de deux célèbres résultats, l'un dû à Ornstein et Weiss et l'autre à Dye montre qu'il n'y a à équivalence orbitale près qu'une seule action de groupe moyennable sur un espace de probabilité standard. Ma principale contribution à travers le présent manuscrit consiste à apporter un éclairage et des preuves dynamiques élémentaires aux classifications obtenues par Giordano, Putnam et Skau (celle sur l'équivalence orbitale susmentionnée ainsi qu'une autre traitant d'une variation nommée équivalence orbitale forte), tant afin de les comprendre sous une autre perspective que pour tenter de les étendre à d'autres contextes. Chemin faisant, je démontrerai également un résultat de complexité Borélienne, à savoir que la relation d'isomorphisme de groupes dénombrables, localement finis et simples et une relation universelle provenant d'une action Borélienne de S_\infty, et nous améliorerons un résultat de Krieger sur la conjugaison des groupes amples
This thesis takes place in the context of topological dynamics, a branch of dynamical systems concerned with the asymptotic qualitative behavior of continuous transformations arising from a group or semigroup action on a usually compact metric space. For example, a classic question might be whether a dynamical system admits recurrent points, i.e. points that will return arbitrarily close to their starting point infinitely often under the dynamics. Often, because of their qualitative and asymptotic nature, these properties do not depend precisely on the system but rather on the orbits of the points, i.e. the positions they will reach. Hence the notion of orbit equivalence at the heart of this thesis, which consists in considering that, after identification of the underlying spaces, two systems whose points all have the same orbits would be "qualitatively the same". In the 1990s, Giordano Putnam and Skau used homological algebra to establish a classification up to orbit equivalence of minimal dynamical systems arising from Z-actions on a Cantor space in terms of both full groups and invariant measures. This result shows in particular that there are non-countably many such different systems up to orbit equivalence, which contrasts quite strongly with the framework of ergodic theory, a very close field concerned with measured dynamical systems, in which the combination of two famous results, one due to Ornstein and Weiss and the other to Dye, shows that there is only one amenable group action on a standard probability space up to orbit equivalence. My main contribution in the present manuscript is to bring an elementary perpective and dynamical proofs to the classifications obtained by Giordano, Putnam and Skau (the one on orbital equivalence mentioned above as well as another one dealing with a variation called strong orbital equivalence), both in order to understand them from another perspective and to try to extend them to other contexts. Along the way, I will also prove a result of Borelian complexity, namely that the isomorphism relation of countable, locally finite and simple groups and a universal relation arising from a Borelian action of S_\infty, and improve a result of Krieger about the conjugation of ample groups
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Craft, Colin N. "Applications of a Model-Theoretic Approach to Borel Equivalence Relations." Thesis, University of North Texas, 2019. https://digital.library.unt.edu/ark:/67531/metadc1538768/.

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The study of Borel equivalence relations on Polish spaces has become a major area of focus within descriptive set theory. Primarily, work in this area has been carried out using the standard methods of descriptive set theory. In this work, however, we develop a model-theoretic framework suitable for the study of Borel equivalence relations, introducing a class of objects we call Borel structurings. We then use these structurings to examine conditions under which marker sets for Borel equivalence relations can be concluded to exist or not exist, as well as investigating to what extent the Compactness Theorem from first-order logic continues to hold for Borel structurings.
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Cotton, Michael R. "Abelian Group Actions and Hypersmooth Equivalence Relations." Thesis, University of North Texas, 2019. https://digital.library.unt.edu/ark:/67531/metadc1505289/.

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We show that any Borel action on a standard Borel space of a group which is topologically isomorphic to the sum of a countable abelian group with a countable sum of lines and circles induces an orbit equivalence relation which is hypersmooth. We also show that any Borel action of a second countable locally compact abelian group on a standard Borel space induces an orbit equivalence relation which is essentially hyperfinite, generalizing a result of Gao and Jackson for the countable abelian groups.
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Hart, Robert. "A Non-commutative *-algebra of Borel Functions." Thèse, Université d'Ottawa / University of Ottawa, 2012. http://hdl.handle.net/10393/23235.

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To the pair (E,c), where E is a countable Borel equivalence relation on a standard Borel space (X,A) and c a normalized Borel T-valued 2-cocycle on E, we associate a sequentially weakly closed Borel *-algebra Br*(E,c), contained in the bounded linear operators on L^2(E). Associated to Br*(E,c) is a natural (Borel) Cartan subalgebra (Definition 6.4.10) L(Bo(X)) isomorphic to the bounded Borel functions on X. Then L(Bo(X)) and its normalizer (the set of the unitaries u in Br*(E,c) such that u*fu in L(Bo(X)), f in L(Bo(X))) countably generates the Borel *-algebra Br*(E,c). In this thesis, we study Br*(E,c) and in particular prove that: i) If E is smooth, then Br*(E,c) is a type I Borel *-algebra (Definition 6.3.10). ii) If E is a hyperfinite, then Br*(E,c) is a Borel AF-algebra (Definition 7.5.1). iii) Generalizing Kumjian's definition, we define a Borel twist G over E and its associated sequentially closed Borel *-algebra Br*(G). iv) Let a Borel Cartan pair (B, Bo) denote a sequentially closed Borel *-algebra B with a Borel Cartan subalgebra Bo, where B is countably Bo-generated. Generalizing Feldman-Moore's result, we prove that any pair (B, Bo) can be realized uniquely as a pair (Br*(E,c), L(Bo(X))). Moreover, we show that the pair (Br*(E,c), L(Bo(X))) is a complete invariant of the countable Borel equivalence relation E. v) We prove a Krieger type theorem, by showing that two aperiodic hyperfinite countable equivalence relations are isomorphic if and only if their associated Borel *-algebras Br*(E1) and Br*(E2) are isomorphic.
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Martin, Tiffani L. "Does Stimulus Complexity Affect Acquisition of Conditional Discriminations and the Emergence of Derived Relations?" Thesis, University of North Texas, 2009. https://digital.library.unt.edu/ark:/67531/metadc12160/.

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Despite the central importance of conditional discriminations to the derivation of equivalence relations, there is little research relating the dynamics of conditional discrimination learning to the derivation of equivalence relations. Prior research has shown that conditional discriminations with simple sample and comparison stimuli are acquired faster than conditional discriminations with complex sample and comparison stimuli. This study attempted to replicate these earlier results and extend them by attempting to relate conditional discrimination learning to equivalence relations. Each of four adult humans learned four, four-choice conditional discriminations (simple-simple, simple-complex, complex-simple, and complex-complex) and were tested to see if equivalence relations had developed. The results confirm earlier findings showing acquisition to be facilitated with simple stimuli and retarded with complex stimuli. There was no difference in outcomes on equivalence tests, however. The results are in implicit agreement with Sidman's theory of stimulus equivalence.
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Sofronidis, Nikolaos Efstathiou. "Topics in descriptive set theory related to equivalence relations, complex borel and analytic sets." Thesis, 1999. https://thesis.library.caltech.edu/10021/1/Sofronidis_NE_1999.pdf.

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The purpose of this doctoral dissertation is first to show that certain kinds of invariants for measures, self-adjoint and unitary operators are as far from complete as possible and second to give new natural examples of complex Borel and analytic sets originating from Analysis and Geometry.

The dissertation is divided in two parts.

In the first part we prove that the measure equivalence relation and certain of its most characteristic subequivalence relations are generically S∞- ergodic and unitary conjugacy of self-adjoint and unitary operators is generically turbulent.

In the second part we prove that for any 0 ≤ α < ∞, the set of entire functions whose order is equal to α is ∏03-complete and the set of all sequences of entire functions whose orders converge to α is ∏05-complete. We also prove that given any line in the plane and any cardinal number 1 ≤ n ≤ N0, the set of continuous paths in the plane tracing curves which admit at least n tangents parallel to the given line is Σ11-complete and the set of differentiable paths of class C2 in the plane admitting a canonical parameter in [0,1] and tracing curves which have at least n vertices is also Σ11-complete.

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Uzcátegui, Carlos. "Smooth sets for borel equivalence relations and the covering property for σ-ideals of compact sets." Thesis, 1990. https://thesis.library.caltech.edu/8783/2/Uzcategui_c_1990.pdf.

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This thesis is divided into three chapters. In the first chapter we study the smooth sets with respect to a Borel equivalence realtion E on a Polish space X. The collection of smooth sets forms σ-ideal. We think of smooth sets as analogs of countable sets and we show that an analog of the perfect set theorem for Σ11 sets holds in the context of smooth sets. We also show that the collection of Σ11 smooth sets is ∏11 on the codes. The analogs of thin sets are called sparse sets. We prove that there is a largest ∏11 sparse set and we give a characterization of it. We show that in L there is a ∏11 sparse set which is not smooth. These results are analogs of the results known for the ideal of countable sets, but it remains open to determine if large cardinal axioms imply that ∏11 sparse sets are smooth. Some more specific results are proved for the case of a countable Borel equivalence relation. We also study I(E), the σ-ideal of closed E-smooth sets. Among other things we prove that E is smooth iff I(E) is Borel.

In chapter 2 we study σ-ideals of compact sets. We are interested in the relationship between some descriptive set theoretic properties like thinness, strong calibration and the covering property. We also study products of σ-ideals from the same point of view. In chapter 3 we show that if a σ-ideal I has the covering property (which is an abstract version of the perfect set theorem for Σ11 sets), then there is a largest ∏11 set in Iint (i.e., every closed subset of it is in I). For σ-ideals on 2ω we present a characterization of this set in a similar way as for C1, the largest thin ∏11 set. As a corollary we get that if there are only countable many reals in L, then the covering property holds for Σ12 sets.

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Doucha, Michal. "Forcing, deskriptivní teorie množin, analýza." Doctoral thesis, 2013. http://www.nusl.cz/ntk/nusl-329275.

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The dissertation thesis consists of two thematic parts. The first part, i.e. chapters 2, 3 and 4, contains results concerning the topic of a new book of the supervisor and coauthors V. Kanovei and M. Sabok "Canonical Ramsey Theory on Polish Spaces". In Chapter 2, there is proved a canonization of all equivalence relations Borel reducible to equivalences definable by analytic P-ideals for the Silver ideal. Moreover, it investigates and classifies sube- quivalences of the equivalence relation E0. In Chapter 3, there is proved a canonization of all equivalence relations Borel reducible to equivalences de- finable by Fσ P-ideals for the Laver ideal and in Chapter 4, we prove the canonization for all analytic equivalence relations for the ideal derived from the Carlson-Simpson (Dual Ramsey) theorem. The second part, consisting of Chapter 5, deals with the existence of universal and ultrahomogeneous Polish metric structures. For instance, we construct a universal Polish metric space which is moreover equipped with countably many closed relations or with a Lipschitz function to an arbitrarily chosen Polish metric space. This work can be considered as an extension of the result of P. Urysohn who constructed a universal and ultrahomogeneous Polish metric space.
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Books on the topic "Borel complexity of equivalence relations"

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Borel equivalence relations: Structure and classification. Providence, R.I: American Mathematical Society, 2008.

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Hjorth, Greg. Rigidity theorems for actions of product groups and countable Borel equivalence relations. Providence, RI: American Mathematical Society, 2005.

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Geometric Set Theory. American Mathematical Society, 2020.

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Book chapters on the topic "Borel complexity of equivalence relations"

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Kanovei, Vladimir. "Borel ideals." In Borel Equivalence Relations, 41–50. Providence, Rhode Island: American Mathematical Society, 2008. http://dx.doi.org/10.1090/ulect/044/04.

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Hjorth, Greg. "Borel Equivalence Relations." In Handbook of Set Theory, 297–332. Dordrecht: Springer Netherlands, 2009. http://dx.doi.org/10.1007/978-1-4020-5764-9_5.

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Kanovei, Vladimir. "Hyperfinite equivalence relations." In Borel Equivalence Relations, 95–106. Providence, Rhode Island: American Mathematical Society, 2008. http://dx.doi.org/10.1090/ulect/044/09.

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Kanovei, Vladimir. "Summable equivalence relations." In Borel Equivalence Relations, 181–90. Providence, Rhode Island: American Mathematical Society, 2008. http://dx.doi.org/10.1090/ulect/044/16.

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Kanovei, Vladimir. "Pinned equivalence relations." In Borel Equivalence Relations, 203–9. Providence, Rhode Island: American Mathematical Society, 2008. http://dx.doi.org/10.1090/ulect/044/18.

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Kanovei, Vladimir. "Reduction of Borel equivalence relations to Borel ideals." In Borel Equivalence Relations, 211–21. Providence, Rhode Island: American Mathematical Society, 2008. http://dx.doi.org/10.1090/ulect/044/19.

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Kanovei, Vladimir. "Introduction." In Borel Equivalence Relations, 1–5. Providence, Rhode Island: American Mathematical Society, 2008. http://dx.doi.org/10.1090/ulect/044/01.

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Kanovei, Vladimir. "Descriptive set theoretic background." In Borel Equivalence Relations, 7–18. Providence, Rhode Island: American Mathematical Society, 2008. http://dx.doi.org/10.1090/ulect/044/02.

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Kanovei, Vladimir. "Some theorems of descriptive set theory." In Borel Equivalence Relations, 19–39. Providence, Rhode Island: American Mathematical Society, 2008. http://dx.doi.org/10.1090/ulect/044/03.

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Kanovei, Vladimir. "Introduction to equivalence relations." In Borel Equivalence Relations, 51–61. Providence, Rhode Island: American Mathematical Society, 2008. http://dx.doi.org/10.1090/ulect/044/05.

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Conference papers on the topic "Borel complexity of equivalence relations"

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Hjorth, Greg. "Countable Borel equivalence relations, Borel reducibility, and orbit equivalence." In 10th Asian Logic Conference. WORLD SCIENTIFIC, 2009. http://dx.doi.org/10.1142/9789814293020_0007.

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