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65
The classification of hypersmooth Borel equivalence relations
 J. Amer. Math. Soc
, 1997
"... This paper is a contribution to the study of Borel equivalence relations in standard Borel spaces, i.e., Polish spaces equipped with their Borel structure. A class of such equivalence relations which has received particular attention is the class of hyperfinite Borel equivalence relations. These can ..."
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Cited by 38 (4 self)
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This paper is a contribution to the study of Borel equivalence relations in standard Borel spaces, i.e., Polish spaces equipped with their Borel structure. A class of such equivalence relations which has received particular attention is the class of hyperfinite Borel equivalence relations. These can be defined as the increasing unions of sequences of Borel equivalence relations all of whose equivalence classes are finite or, as it turns out, equivalently those induced by the orbits of a single Borel automorphism. Hyperfinite equivalence relations have been classified in [DJK], under two notions of equivalence, Borel bireducibility, and Borel isomorphism. An equivalence relation E on X is Borel reducible to an equivalence relation F on Y if there is a Borel map f: X → Y with xEy ⇔ f(x)Ff(y). We write then E ≤ F. If E ≤ Fand F ≤ E we say that E,F are Borel bireducible, in symbols E ≈ ∗ F.When E ≈ ∗ Fthe quotient spaces X/E, Y/F have the same “effective ” or “definable ” cardinality. We say that E,F are Borel isomorphic if there exists a Borel bijection f: X → Y with xEy ⇔ f(x)Ff(y). Below we denote by E0,Et the equivalence relations on the Cantor space 2 N given by: xE0y ⇔
Degree spectra and computable dimension in algebraic structures
 Annals of Pure and Applied Logic 115 (2002
, 2002
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The classification problem for torsionfree abelian groups of finite rank
 J. Amer. Math. Soc
, 2001
"... In 1937, Baer [5] introduced the notion of the type of an element in a torsionfree abelian group and showed that this notion provided a complete invariant for the classification problem for torsionfree abelian groups of rank 1. Since then, despite the efforts of such mathematicians as Kurosh [23] ..."
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Cited by 24 (8 self)
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In 1937, Baer [5] introduced the notion of the type of an element in a torsionfree abelian group and showed that this notion provided a complete invariant for the classification problem for torsionfree abelian groups of rank 1. Since then, despite the efforts of such mathematicians as Kurosh [23] and Malcev [25], no satisfactory
On the Complexity of the Isomorphism Relation for Finitely Generated Groups
, 1998
"... Working within the framework of descriptive set theory, we show that the isomorphism relation for finitely generated groups is a universal essentially countable Borel equivalence relation. We also prove the corresponding result for the conjugacy relation for subgroups of the free group on two genera ..."
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Cited by 20 (11 self)
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Working within the framework of descriptive set theory, we show that the isomorphism relation for finitely generated groups is a universal essentially countable Borel equivalence relation. We also prove the corresponding result for the conjugacy relation for subgroups of the free group on two generators. The proofs are grouptheoretic, and we refer to descriptive set theory only for the relevant definitions and for motivation for the results. Introduction Given a class K of structures for a fixed first order language L, one may ask what kinds of complete invariants can be used to classify the elements of K up to isomorphism. For those classes consisting of the countable models of some L ! 1 ;! sentence, Friedman and Stanley [FS] proposed to use the methods of descriptive set theory to study their possible invariants and defined the notion of Borel reducibility between such classes of structures. In [HK], Hjorth and Kechris continued this study and situated it within the general the...
Superrigidity and countable Borel equivalence relations
 Annals Pure Appl. Logic
"... Introduction. These notes are based upon a daylong lecture workshop presented by Simon ..."
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Cited by 14 (6 self)
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Introduction. These notes are based upon a daylong lecture workshop presented by Simon
Cofinal families of Borel equivalence relations and quasiorders
 J. Symbolic Logic
"... Abstract. Families of Borel equivalence relations and quasiorders that are coÞnal with respect to the Borel reducibility ordering, B, are constructed. There is an analytic ideal on ù generating a complete analytic equivalence relation and any Borel equivalence relation reduces to one generated by a ..."
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Cited by 13 (2 self)
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Abstract. Families of Borel equivalence relations and quasiorders that are coÞnal with respect to the Borel reducibility ordering, B, are constructed. There is an analytic ideal on ù 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 Kó complete. x1. Introduction. For R and S binary relations on Polish spaces X and Y, respectively, one writes R B S and says thatR Borel reduces to S, if there is a Borel functionf: X! Y such that xRy () f(x)Sf(y) (i » (ff)1(S) = R). Usually this ordering has been studied when R and S both are equivalence relations, but recently it has turned out that the study of quasiorders, i.e., reßexive, transitive
The completeness of the isomorphism relation for countable Boolean algebras
 Trans. Amer. Math. Soc
"... Abstract. We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen’s classification of countable Boolean algebras is o ..."
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Cited by 12 (2 self)
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Abstract. We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen’s classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an essential way. We also give a stronger form of the Vaught conjecture for Boolean algebras which states that, for any complete firstorder theory of Boolean algebras that has more than one countable model up to isomorphism, the class of countable models for the theory is Borel complete. The results are applied to settle many other classification problems related to countable Boolean algebras and separable Boolean spaces. In particular, we will show that the following equivalence relations are Borel complete: the translation equivalence between closed subsets of the Cantor space, the isomorphism relation between ideals of the countable atomless Boolean algebra, the conjugacy equivalence of the autohomeomorphisms of the Cantor space, etc. Another corollary of our results is the Borel completeness of the commutative AF C ∗algebras, which in turn gives rise to similar results for Bratteli diagrams and dimension groups. 1.
The classification problem for plocal torsionfree abelian groups of finite rank
, 2002
"... Let n ≥ 3. We prove that if p != q are distinct primes, then the classification problems for plocal and qlocal torsionfree abelian groups of rank n are incomparable with respect to Borel reducibility. ..."
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Cited by 11 (4 self)
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Let n ≥ 3. We prove that if p != q are distinct primes, then the classification problems for plocal and qlocal torsionfree abelian groups of rank n are incomparable with respect to Borel reducibility.
Polish group actions: dichotomies and generalized elementary embeddings
 J. Amer. Math. Soc
, 1997
"... The results in this paper involve two different topics in the descriptive theory of Polish group actions. The book BeckerKechris [6] is an introduction to that theory. Our two topics—and two collections of theorems—are rather unrelated, but the proofs for both topics are essentially the same. ..."
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Cited by 10 (0 self)
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The results in this paper involve two different topics in the descriptive theory of Polish group actions. The book BeckerKechris [6] is an introduction to that theory. Our two topics—and two collections of theorems—are rather unrelated, but the proofs for both topics are essentially the same.
The complexity of classifying separable Banach spaces up to isomorphism
, 2006
"... It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, i.e., that any analytic equivalence relation Borel reduces to it. Thus, separable Banach spaces up to isomorphism provide complete invariants for a great number of mathematica ..."
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Cited by 9 (2 self)
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It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, i.e., that any analytic equivalence relation Borel reduces to it. Thus, separable Banach spaces up to isomorphism provide complete invariants for a great number of mathematical structures up to their corresponding notion of isomorphism. The same is shown to hold for (1) complete separable metric spaces up to uniform homeomorphism, (2) separable Banach spaces up to Lipschitz isomorphism, and (3) up to (complemented) biembeddability, (4) Polish groups up to topological isomorphism, and (5) Schauder bases up to permutative equivalence. Some of the constructions rely on methods recently developed by S. Argyros and P. Dodos.