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37
Countable Borel Equivalence Relations
 J. Math. Logic
"... This paper is a contribution to a new direction in descriptive set theory that is being extensively pursued over the last decade or so. It deals with the development of the theory of definable actions of Polish groups, the structure and classification of their orbit spaces, and the closely related s ..."
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Cited by 44 (7 self)
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This paper is a contribution to a new direction in descriptive set theory that is being extensively pursued over the last decade or so. It deals with the development of the theory of definable actions of Polish groups, the structure and classification of their orbit spaces, and the closely related study of definable equivalence relations. This study is motivated by basic foundational questions, like understanding the nature of complete classification of mathematical objects, up to some notion of equivalence, by invariants, and creating a mathematical framework for measuring the complexity of such classification problems. (For an extensive discussion of these matters, see, e.g., Hjorth [00], Kechris [99, 00a].) This theory is developed within the context of descriptive set theory, which provides the basic underlying concepts and methods. On the other hand, in view of its broad scope, there are natural interactions of it with other areas of mathematics, such as model theory, recursion theory, the theory of topological groups and their representations, topological dynamics, ergodic theory, and operator algebras
Turbulence, amalgamation, and generic automorphisms of homogeneous structures
 Proc. London Math. Soc
, 2004
"... Abstract. We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological Rokhlin property). We then characterize when an automorphism group ad ..."
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Cited by 34 (6 self)
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Abstract. We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological Rokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class (answering a question of Truss) and apply this to show that the homeomorphism group of the Cantor space has a comeager conjugacy class (answering a question of AkinHurleyKennedy). Finally, we study Polish groups that admit comeager conjugacy classes in any dimension (in which case the groups are said to admit ample generics). We show that Polish groups with ample generics have the small index property (generalizing results of HodgesHodkinsonLascarShelah) and arbitrary homomorphisms from such groups into separable groups are automatically continuous. Moreover, in the case of oligomorphic permutation groups, they have uncountable cofinality and the Bergman property. These results in particular apply to automorphism groups of many ωstable, ℵ0categorical structures and of the random graph. In this connection, we also show that the infinite symmetric group S ∞ has a unique nontrivial separable group topology. For several interesting groups we also establish Serre’s properties (FH) and (FA). 1.
Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations
"... This paper is a contribution to the theory of countable Borel equivalence relations on standard Borel spaces. As usual, by a standard Borel space we mean a Polish (complete separable metric) space equipped with its #algebra of Borel sets. An equivalence relation E on a standard Borel space X is Bor ..."
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Cited by 25 (6 self)
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This paper is a contribution to the theory of countable Borel equivalence relations on standard Borel spaces. As usual, by a standard Borel space we mean a Polish (complete separable metric) space equipped with its #algebra of Borel sets. An equivalence relation E on a standard Borel space X is Borel if it is a Borel subset of X². Given two
Complete analytic equivalence relations
 Trans. Amer. Math. Soc
"... Abstract. We prove that various concrete analytic equivalence relations arising in model theory or analysis are complete, i.e. maximum in the Borel reducibility ordering. The proofs use some general results concerning the wider class of analytic quasiorders. ..."
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Cited by 12 (2 self)
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Abstract. We prove that various concrete analytic equivalence relations arising in model theory or analysis are complete, i.e. maximum in the Borel reducibility ordering. The proofs use some general results concerning the wider class of analytic quasiorders.
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 10 (1 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.
Actions of Polish Groups and Classification Problems
 London Math. Soc. Lecture Note Series
"... Analysis, (1969), SpringerVerlag, New York. Hjorth, G. [97]. On the isomorphism problem for measure preserving transformations, (April 1997), preprint. 67 68 References ... ..."
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Cited by 9 (4 self)
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Analysis, (1969), SpringerVerlag, New York. Hjorth, G. [97]. On the isomorphism problem for measure preserving transformations, (April 1997), preprint. 67 68 References ...
Topologies on the group of Borel automorphisms of a standard Borel space, preprint
, 2003
"... The paper is devoted to the study of topologies on the group Aut(X, B) of all Borel automorphisms of a standard Borel space (X, B). Several topologies are introduced and all possible relations between them are found. One of these topologies, τ, is a direct analogue of the uniform topology widely use ..."
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Cited by 8 (5 self)
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The paper is devoted to the study of topologies on the group Aut(X, B) of all Borel automorphisms of a standard Borel space (X, B). Several topologies are introduced and all possible relations between them are found. One of these topologies, τ, is a direct analogue of the uniform topology widely used in ergodic theory. We consider the most natural subsets of Aut(X, B) and find their closures. In particular, we describe closures of subsets formed by odometers, periodic, aperiodic, incompressible, and smooth automorphisms with respect to the defined topologies. It is proved that the set of periodic Borel automorphisms is dense in Aut(X, B) (Rokhlin lemma) with respect to τ. It is shown that the τclosure of odometers (and of rank 1 Borel automorphisms) coincides with the set of all aperiodic automorphisms. For every aperiodic automorphism T ∈ Aut(X, B), the concept of a BorelBratteli diagram is defined and studied. It is proved that every aperiodic Borel automorphism T is isomorphic to the Vershik transformation acting on the space of infinite paths of an ordered BorelBratteli diagram. Several applications of this result are given.
Polish Metric Spaces: Their Classification and Isometry Groups
 Bull. Symbolic Logic
"... this paper. For example, it is well known that E#
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Cited by 8 (0 self)
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this paper. For example, it is well known that E# <B F 2 . From the realization of E# as the isomorphism relation among locally finite trees, it is not hard to see that E# is Borel reducible to the isometry of connected locally compact Polish metric spaces. Again results on their isometry groups motivated our conjecture that the isometry of connected locally compact Polish metric spaces is Borel bireducible with E# . This has been confirmed by Hjorth
Prospects for mathematical logic in the twentyfirst century
 BULLETIN OF SYMBOLIC LOGIC
, 2002
"... The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently. ..."
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Cited by 8 (0 self)
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The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
Turbulence, representations, and tracepreserving actions
 Proc. Lond. Math. Soc
, 2010
"... Abstract. We establish criteria for turbulence in certain spaces of C ∗algebra representations and apply this to the problem of nonclassifiability by countable structures for group actions on a standard atomless probability space (X, µ) and on the hyperfinite II1 factor R. We also prove that the co ..."
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Cited by 7 (0 self)
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Abstract. We establish criteria for turbulence in certain spaces of C ∗algebra representations and apply this to the problem of nonclassifiability by countable structures for group actions on a standard atomless probability space (X, µ) and on the hyperfinite II1 factor R. We also prove that the conjugacy action on the space of free actions of a countably infinite amenable group on R is turbulent, and that the conjugacy action on the space of ergodic measurepreserving flows on (X, µ) is generically turbulent. 1.