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

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Published: 13 April 2024

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1

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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

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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10

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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11

Kechris, Alexander S. "Amenable versus hyperfinite Borel equivalence relations." Journal of Symbolic Logic 58, no. 3 (September 1993): 894–907. http://dx.doi.org/10.2307/2275102.

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LetXbe a standard Borel space (i.e., a Polish space with the associated Borel structure), and letEbe acountableBorel equivalence relation onX, i.e., a Borel equivalence relationEfor which every equivalence class [x]Eis countable. By a result of Feldman-Moore [FM],Eis induced by the orbits of a Borel action of a countable groupGonX.The structure of general countable Borel equivalence relations is very little understood. However, a lot is known for the particularly important subclass consisting of hyperfinite relations. A countable Borel equivalence relation is calledhyperfiniteif it is induced by a Borel ℤ-action, i.e., by the orbits of a single Borel automorphism. Such relations are studied and classified in [DJK] (see also the references contained therein). It is shown in Ornstein-Weiss [OW] and Connes-Feldman-Weiss [CFW] that for every Borel equivalence relationEinduced by a Borel action of a countable amenable groupGonXand for every (Borel) probability measure μ onX, there is a Borel invariant setY⊆Xwith μ(Y) = 1 such thatE↾Y(= the restriction ofEtoY) is hyperfinite. (Recall that a countable group G isamenableif it carries a finitely additive translation invariant probability measure defined on all its subsets.) Motivated by this result, Weiss [W2] raised the question of whether everyEinduced by a Borel action of a countable amenable group is hyperfinite. Later on Weiss (personal communication) showed that this is true forG= ℤn. However, the problem is still open even for abelianG. Our main purpose here is to provide a weaker affirmative answer for general amenableG(and more—see below). We need a definition first. Given two standard Borel spacesX, Y, auniversally measurableisomorphism betweenXandYis a bijection ƒ:X→Ysuch that both ƒ, ƒ-1are universally measurable. (As usual, a mapg:Z→W, withZandWstandard Borel spaces, is calleduniversally measurableif it is μ-measurable for every probability measure μ onZ.) Notice now that to assert that a countable Borel equivalence relation onXis hyperfinite is trivially equivalent to saying that there is a standard Borel spaceYand a hyperfinite Borel equivalence relationFonY, which isBorelisomorphic toE, i.e., there is a Borel bijection ƒ:X→YwithxEy⇔ ƒ(x)Fƒ(y). We have the following theorem.

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12

Hjorth, Greg. "Bi-Borel reducibility of essentially countable Borel equivalence relations." Journal of Symbolic Logic 70, no. 3 (September 2005): 979–92. http://dx.doi.org/10.2178/jsl/1122038924.

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This note answers a questions from [2] by showing that considered up to Borel reducibility, there are more essentially countable Borel equivalence relations than countable Borel equivalence relations. Namely:Theorem 0.1. There is an essentially countable Borel equivalence relation E such that for no countable Borel equivalence relation F (on a standard Borel space) do we haveThe proof of the result is short. It does however require an extensive rear guard campaign to extract from the techniques of [1] the followingMessy Fact 0.2. There are countable Borel equivalence relationssuch that:(i) eachExis defined on a standard Borel probability space (Xx, μx); each Ex is μx-invariant and μx-ergodic;(ii) forx1 ≠ x2 and A μxι -conull, we haveExι/Anot Borel reducible toEx2;(iii) if f: Xx → Xxis a measurable reduction ofExto itself then(iv)is a standard Borel space on which the projection functionis Borel and the equivalence relation Ê given byif and only ifx = x′ andzExz′ is Borel;(V)is Borel.We first prove the theorem granted this messy fact. We then prove the fact.(iv) and (v) are messy and unpleasant to state precisely, but are intended to express the idea that we have an effective parameterization of countable Borel equivalence relations by points in a standard Borel space. Examples along these lines appear already in the Adams-Kechris constructions; the new feature is (iii).Simon Thomas has pointed out to me that in light of theorem 4.4 [5] the Gefter-Golodets examples of section 5 [5] also satisfy the conclusion of 0.2.

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13

Hjorth, Greg, and Alexander S. Kechris. "New Dichotomies for Borel Equivalence Relations." Bulletin of Symbolic Logic 3, no. 3 (September 1997): 329–46. http://dx.doi.org/10.2307/421148.

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We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.§1. Introduction. For X a Polish (i.e., separable, completely metrizable) space and E a Borel equivalence relation on X, a (complete) classification of X up to E-equivalence consists of finding a set of invariants I and a map c : X → I such that xEy ⇔ c(x) = c(y). To be of any value we would expect I and c to be “explicit” or “definable”. The theory of Borel equivalence relations investigates the nature of possible invariants and provides a hierarchy of notions of classification.The following partial (pre-)ordering is fundamental in organizing this study. Given equivalence relations E and F on X and Y, resp., we say that E can be Borel reduced to F, in symbolsif there is a Borel map f : X → Y with xEy ⇔ f(x)Ff(y). Then if is an embedding of X/E into Y/F, which is “Borel” (in the sense that it has a Borel lifting).Intuitively, E ≤BF might be interpreted in any one of the following ways:(i) The classi.cation problem for E is simpler than (or can be reduced to) that of F: any invariants for F work as well for E (after composing by an f as above).(ii) One can classify E by using as invariants F-equivalence classes.(iii) The quotient space X/E has “Borel cardinality” less than or equal to that of Y/F, in the sense that there is a “Borel” embedding of X/E into Y/F.

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14

DOUCHA, MICHAL. "Fσ EQUIVALENCE RELATIONS AND LAVER FORCING." Journal of Symbolic Logic 79, no. 2 (June 2014): 644–53. http://dx.doi.org/10.1017/jsl.2013.32.

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AbstractFollowing the topic of the book Canonical Ramsey Theory on Polish Spaces by V. Kanovei, M. Sabok, and J. Zapletal we study Borel equivalences on Laver trees. We prove that equivalence relations Borel reducible to an equivalence relation on 2ω given by some FσP-ideal on ω can be canonized to the full equivalence relation or to the identity relation.This has several consequences, e.g., Silver type dichotomy for the Laver ideal and equivalences Borel reducible to equivalence relations given by FσP-ideals.

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15

Bartoš, Adam. "Borel complexity up to the equivalence." Topology and its Applications 270 (February 2020): 107042. http://dx.doi.org/10.1016/j.topol.2019.107042.

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16

Hjorth, Greg, and André Nies. "Borel structures and Borel theories." Journal of Symbolic Logic 76, no. 2 (June 2011): 461–76. http://dx.doi.org/10.2178/jsl/1305810759.

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AbstractWe show that there is a complete, consistent Borel theory which has no “Borel model” in the following strong sense: There is no structure satisfying the theory for which the elements of the structure are equivalence classes under some Borel equivalence relation and the interpretations of the relations and function symbols are uniformly Borel.We also investigate Borel isomorphisms between Borel structures.

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17

Vlitas, Dimitris. "Canonical Borel equivalence relations onRn." Topology and its Applications 175 (September 2014): 29–37. http://dx.doi.org/10.1016/j.topol.2014.06.009.

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18

Hjorth, Greg. "Borel equivalence relations which are highly unfree." Journal of Symbolic Logic 73, no. 4 (December 2008): 1271–77. http://dx.doi.org/10.2178/jsl/1230396917.

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19

Marks, Andrew S. "Uniformity, universality, and computability theory." Journal of Mathematical Logic 17, no. 01 (March 28, 2017): 1750003. http://dx.doi.org/10.1142/s0219061317500039.

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We prove a number of results motivated by global questions of uniformity in computabi- lity theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups. We begin by investigating the notion of uniform universality, first proposed by Montalbán, Reimann and Slaman. This notion is a strengthened form of a countable Borel equivalence relation being universal, which we conjecture is equivalent to the usual notion. With this additional uniformity hypothesis, we can answer many questions concerning how countable groups, probability measures, the subset relation, and increasing unions interact with universality. For many natural classes of countable Borel equivalence relations, we can also classify exactly which are uniformly universal. We also show the existence of refinements of Martin’s ultrafilter on Turing invariant Borel sets to the invariant Borel sets of equivalence relations that are much finer than Turing equivalence. For example, we construct such an ultrafilter for the orbit equivalence relation of the shift action of the free group on countably many generators. These ultrafilters imply a number of structural properties for these equivalence relations.

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20

THOMAS, SIMON. "PROPERTY τ AND COUNTABLE BOREL EQUIVALENCE RELATIONS." Journal of Mathematical Logic 07, no. 01 (June 2007): 1–34. http://dx.doi.org/10.1142/s0219061307000603.

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We prove Borel superrigidity results for suitably chosen actions of groups of the form SL2(ℤ[1/p1, … , 1/pt]), where {p1, …, pt} is a finite nonempty set of primes, and present a number of applications to the theory of countable Borel equivalence relations. In particular, for each prime q, we prove that the orbit equivalence relations arising from the natural actions of SL2(ℤ[1/q]) on the projective lines ℚp ∪ {∞}, p ≠ q, over the various p-adic fields are pairwise incomparable with respect to Borel reducibility.

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21

Louveau, Alain, and Boban Veličkovi{ć. "A note on Borel equivalence relations." Proceedings of the American Mathematical Society 120, no. 1 (January 1, 1994): 255. http://dx.doi.org/10.1090/s0002-9939-1994-1169042-2.

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22

Thomas, Simon. "Superrigidity and countable Borel equivalence relations." Annals of Pure and Applied Logic 120, no. 1-3 (April 2003): 237–62. http://dx.doi.org/10.1016/s0168-0072(02)00068-4.

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23

CHAN, WILLIAM. "EQUIVALENCE RELATIONS WHICH ARE BOREL SOMEWHERE." Journal of Symbolic Logic 82, no. 3 (September 2017): 893–930. http://dx.doi.org/10.1017/jsl.2017.22.

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AbstractThe following will be shown: Let I be a σ-ideal on a Polish space X so that the associated forcing of I+${\bf{\Delta }}_1^1$ sets ordered by ⊆ is a proper forcing. Let E be a ${\bf{\Sigma }}_1^1$ or a ${\bf{\Pi }}_1^1$ equivalence relation on X with all equivalence classes ${\bf{\Delta }}_1^1$. If for all $z \in {H_{{{\left( {{2^{{\aleph _0}}}} \right)}^ + }}}$, z♯ exists, then there exists an I+${\bf{\Delta }}_1^1$ set C ⊆ X such that E ↾ C is a ${\bf{\Delta }}_1^1$ equivalence relation.

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24

ANDEREGG, MARTIN, and PHILIPPE HENRY. "Actions of amenable equivalence relations on CAT(0) fields." Ergodic Theory and Dynamical Systems 34, no. 1 (October 30, 2012): 21–54. http://dx.doi.org/10.1017/etds.2012.122.

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AbstractWe present the general notion of Borel fields of metric spaces and show some properties of such fields. Then we make the study specific to the Borel fields of proper CAT(0) spaces and we show that the standard tools we need behave in a Borel way. We also introduce the notion of the action of an equivalence relation on Borel fields of metric spaces and we obtain a rigidity result for the action of an amenable equivalence relation on a Borel field of proper finite dimensional CAT(0) spaces. This main theorem is inspired by the result obtained by Adams and Ballmann regarding the action of an amenable group on a proper CAT(0) space.

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DING, LONGYUN. "ON EQUIVALENCE RELATIONS GENERATED BY SCHAUDER BASES." Journal of Symbolic Logic 82, no. 4 (December 2017): 1459–81. http://dx.doi.org/10.1017/jsl.2017.67.

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AbstractIn this article, a notion of Schauder equivalence relation ℝℕ/L is introduced, where L is a linear subspace of ℝℕ and the unit vectors form a Schauder basis of L. The main theorem is to show that the following conditions are equivalent:(1) the unit vector basis is boundedly complete;(2) L is a Fσ in ℝℕ;(3) ℝℕ/L is Borel reducible to ℓ∞.We show that any Schauder equivalence relation generalized by a basis of ℓ2 is Borel bireducible to ℝℕ/ℓ2 itself, but it is not true for bases of c0 or ℓ1. Furthermore, among all Schauder equivalence relations generated by sequences in c0, we find the minimum and the maximum elements with respect to Borel reducibility.

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Friedman, Sy-David, and Luca Motto Ros. "Analytic equivalence relations and bi-embeddability." Journal of Symbolic Logic 76, no. 1 (March 2011): 243–66. http://dx.doi.org/10.2178/jsl/1294170999.

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AbstractLouveau and Rosendal [5] have shown that the relation of bi-embeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphism relation, which as an equivalence relation on graphs (or on any class of countable structures consisting of the models of a sentence of ) is far from complete (see [5, 2]).In this article we strengthen the results of [5] by showing that not only does bi-embeddability give rise to analytic equivalence relations which are complete under Borel reducibility, but in fact any analytic equivalence relation is Borel equivalent to such a relation. This result and the techniques introduced answer questions raised in [5] about the comparison between isomorphism and bi-embeddability. Finally, as in [5] our results apply not only to classes of countable structures defined by sentences of , but also to discrete metric or ultrametric Polish spaces, compact metrizable topological spaces and separable Banach spaces, with various notions of embeddability appropriate for these classes, as well as to actions of Polish monoids.

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27

Melleray, Julien. "Dynamical simplices and Borel complexity of orbit equivalence." Israel Journal of Mathematics 236, no. 1 (February 12, 2020): 317–44. http://dx.doi.org/10.1007/s11856-020-1976-1.

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28

Conley, Clinton, and Benjamin Miller. "Measure reducibility of countable Borel equivalence relations." Annals of Mathematics 185, no. 2 (March 1, 2017): 347–402. http://dx.doi.org/10.4007/annals.2017.185.2.1.

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Dougherty, R., S. Jackson, and A. S. Kechris. "The structure of hyperfinite Borel equivalence relations." Transactions of the American Mathematical Society 341, no. 1 (January 1, 1994): 193–225. http://dx.doi.org/10.1090/s0002-9947-1994-1149121-0.

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30

Kechris, Alexander S., and Alain Louveau. "The classification of hypersmooth Borel equivalence relations." Journal of the American Mathematical Society 10, no. 1 (1997): 215–42. http://dx.doi.org/10.1090/s0894-0347-97-00221-x.

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31

Coskey, Samuel, and Scott Schneider. "Cardinal characteristics and countable Borel equivalence relations." Mathematical Logic Quarterly 63, no. 3-4 (October 27, 2017): 211–27. http://dx.doi.org/10.1002/malq.201400111.

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32

Thomas, Simon. "Popa Superrigidity and countable Borel equivalence relations." Annals of Pure and Applied Logic 158, no. 3 (April 2009): 175–89. http://dx.doi.org/10.1016/j.apal.2007.08.003.

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Fokina, Ekaterina B., Sy-David Friedman, and Asger Törnquist. "The effective theory of Borel equivalence relations." Annals of Pure and Applied Logic 161, no. 7 (April 2010): 837–50. http://dx.doi.org/10.1016/j.apal.2009.10.002.

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34

Hjorth, Greg. "Glimm-Effros for coanalytic equivalence relations." Journal of Symbolic Logic 74, no. 2 (June 2009): 402–22. http://dx.doi.org/10.2178/jsl/1243948320.

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35

YIN, ZHI. "EMBEDDINGS OF P(ω)/Fin INTO BOREL EQUIVALENCE RELATIONS BETWEEN ℓp AND ℓq." Journal of Symbolic Logic 80, no. 3 (July 22, 2015): 917–39. http://dx.doi.org/10.1017/jsl.2015.20.

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AbstractWe prove that, for 1 ≤ p < q < ∞, the partially ordered set P(ω)/Fin can be embedded into Borel equivalence relations between ℝω/ℓp and ℝω/ℓq. Since there is an antichain of size continuum in P(ω)/Fin, there are continuum many pairwise incomparable Borel equivalence relations between ℝω/ℓp and ℝω/ℓq.

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36

Hjorth, Greg, and Alexander S. Kechris. "Analytic equivalence relations and Ulm-type classifications." Journal of Symbolic Logic 60, no. 4 (December 1995): 1273–300. http://dx.doi.org/10.2307/2275888.

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Our main goal in this paper is to establish a Glimm-Effros type dichotomy for arbitrary analytic equivalence relations.The original Glimm-Effros dichotomy, established by Effros [Ef], [Ef1], who generalized work of Glimm [G1], asserts that if an Fσ equivalence relation on a Polish space X is induced by the continuous action of a Polish group G on X, then exactly one of the following alternatives holds:(I) Elements of X can be classified up to E-equivalence by “concrete invariants” computable in a reasonably definable way, i.e., there is a Borel function f: X → Y, Y a Polish space, such that xEy ⇔ f(x) = f(y), or else(II) E contains a copy of a canonical equivalence relation which fails to have such a classification, namely the relation xE0y ⇔ ∃n∀m ≥ n(x(n) = y(n)) on the Cantor space 2ω (ω = {0,1,2, …}), i.e., there is a continuous embedding g: 2ω → X such that xE0y ⇔ g(x)Eg(y).Moreover, alternative (II) is equivalent to:(II)′ There exists an E-ergodic, nonatomic probability Borel measure on X, where E-ergodic means that every E-invariant Borel set has measure 0 or 1 and E-nonatomic means that every E-equivalence class has measure 0.

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37

Adams, Scot. "Trees and amenable equivalence relations." Ergodic Theory and Dynamical Systems 10, no. 1 (March 1990): 1–14. http://dx.doi.org/10.1017/s0143385700005368.

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AbstractLet R be a Borel equivalence relation with countable equivalence classes on a measure space M. Intuitively, a ‘treeing’ of R is a measurably-varying way of makin each equivalence class into the vertices of a tree. We make this definition rigorous. We prove that if each equivalence class becomes a tree with polynomial growth, then the equivalence relation is amenable. We prove that if the equivalence relation is finite measure-preserving and amenable, then almost every tree (i.e., equivalence class) must have one or two ends.

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38

Coskey, Samuel. "Ioana's Superrigidity Theorem and Orbit Equivalence Relations." ISRN Algebra 2013 (December 30, 2013): 1–8. http://dx.doi.org/10.1155/2013/387540.

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We give a survey of Adrian Ioana's cocycle superrigidity theorem for profinite actions of Property (T) groups and its applications to ergodic theory and set theory in this expository paper. In addition to a statement and proof of Ioana's theorem, this paper features the following: (i) an introduction to rigidity, including a crash course in Borel cocycles and a summary of some of the best-known superrigidity theorems; (ii) some easy applications of superrigidity, both to ergodic theory (orbit equivalence) and set theory (Borel reducibility); and (iii) a streamlined proof of Simon Thomas's theorem that the classification of torsion-free abelian groups of finite rank is intractable.

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39

Camerlo, Riccardo. "The relation of recursive isomorphism for countable structures." Journal of Symbolic Logic 67, no. 2 (June 2002): 879–95. http://dx.doi.org/10.2178/jsl/1190150114.

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AbstractIt is shown that the relations of recursive isomorphism on countable trees, groups, Boolean algebras, fields and total orderings are universal countable Borel equivalence relations, thus providing a countable analogue of the Borel completeness of the isomorphism relations on these same classes.

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40

Kechris. "A GLIMM-EFFROS DICHOTOMY FOR BOREL EQUIVALENCE RELATIONS." Real Analysis Exchange 16, no. 1 (1990): 23. http://dx.doi.org/10.2307/44153659.

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41

Adams, Scot, and Alexander S. Kechris. "Linear algebraic groups and countable Borel equivalence relations." Journal of the American Mathematical Society 13, no. 4 (June 23, 2000): 909–43. http://dx.doi.org/10.1090/s0894-0347-00-00341-6.

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42

Harrington, L. A., A. S. Kechris, and A. Louveau. "A Glimm-Effros dichotomy for Borel equivalence relations." Journal of the American Mathematical Society 3, no. 4 (1990): 903. http://dx.doi.org/10.1090/s0894-0347-1990-1057041-5.

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43

Hjorth, Greg, and Alexander S. Kechris. "Borel equivalence relations and classifications of countable models." Annals of Pure and Applied Logic 82, no. 3 (December 1996): 221–72. http://dx.doi.org/10.1016/s0168-0072(96)00006-1.

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44

Friedman, Sy-David, and Tapani Hyttinen. "On Borel equivalence relations in generalized Baire space." Archive for Mathematical Logic 51, no. 3-4 (January 20, 2012): 299–304. http://dx.doi.org/10.1007/s00153-011-0266-3.

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45

Zapletal, Jindřich. "Analytic Equivalence Relations and the Forcing Method." Bulletin of Symbolic Logic 19, no. 4 (September 2013): 473–90. http://dx.doi.org/10.1017/s107989860001057x.

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AbstractI describe several ways in which forcing arguments can be used to yield clean and conceptual proofs of nonreducibility, ergodicity and other results in the theory of analytic equivalence relations. In particular, I present simple Borel equivalence relationsE, Fsuch that a natural proof of nonreducibility ofEtoFuses the independence of the Singular Cardinal Hypothesis at ℵω.

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46

Solecki, Sławomir. "Actions of non-compact and non-locally compact Polish groups." Journal of Symbolic Logic 65, no. 4 (December 2000): 1881–94. http://dx.doi.org/10.2307/2695084.

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AbstractWe show that each non-compact Polish group admits a continuous action on a Polish space with non-smooth orbit equivalence relation. We actually construct a free such action. Thus for a Polish group compactness is equivalent to all continuous free actions of this group being smooth. This answers a question of Kechris. We also establish results relating local compactness of the group with its inability to induce orbit equivalence relations not reducible to countable Borel equivalence relations. Generalizing a result of Hjorth, we prove that each non-locally compact, that is, infinite dimensional, separable Banach space has a continuous action on a Polish space with non-Borel orbit equivalence relation, thus showing that this property characterizes non-local compactness among Banach spaces.

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47

Louveau, Alain, Jack H. Silver, John P. Burgess, L. Harrington, R. Sami, Maurice Boffa, Dirk van Dalen, et al. "Counting the Number of Equivalence Classes of Borel and Coanalytic Equivalence Relations." Journal of Symbolic Logic 52, no. 3 (September 1987): 869. http://dx.doi.org/10.2307/2274373.

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48

Ando, Hiroshi, and Yasumichi Matsuzawa. "On Borel equivalence relations related to self-adjoint operators." Journal of Operator Theory 74, no. 1 (July 2015): 183. http://dx.doi.org/10.7900/jot.2014may24.2030.

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49

Smythe, Iian B. "Borel equivalence relations in the space of bounded operators." Fundamenta Mathematicae 237, no. 1 (2017): 31–45. http://dx.doi.org/10.4064/fm116-9-2016.

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50

Kanovei, Vladimir G., and M. Reeken. "Some new results on Borel irreducibility of equivalence relations." Izvestiya: Mathematics 67, no. 1 (February 28, 2003): 55–76. http://dx.doi.org/10.1070/im2003v067n01abeh000418.

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