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65
The isometry group of the Urysohn space as a Lévy Group
, 2005
"... We prove that the isometry group Iso (U) of the universal Urysohn metric space U equipped with the natural Polish topology is a Lévy group in the sense of Gromov and Milman, that is, admits an approximating chain of compact (in fact, finite) subgroups, exhibiting the phenomenon of concentration of m ..."
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Cited by 11 (3 self)
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We prove that the isometry group Iso (U) of the universal Urysohn metric space U equipped with the natural Polish topology is a Lévy group in the sense of Gromov and Milman, that is, admits an approximating chain of compact (in fact, finite) subgroups, exhibiting the phenomenon of concentration of measure. This strengthens an earlier result by Vershik stating that Iso (U) has a dense locally finite subgroup. We propose a reformulation of Connes’ Embedding Conjecture as an approximationtype statement about the unitary group U(ℓ²), and show that in this form the conjecture makes sense also for Iso(U).
Countable dense homogeneity of definable spaces
 PROC. AMER. MATH. SOC
, 2004
"... We investigate which definable separable metric spaces are countable dense homogeneous (CDH). We prove that a Borel CDH space is completely metrizable and give a complete list of zerodimensional Borel CDH spaces. We also show that for a Borel X ⊆ 2 ω the following are equivalent: (1) X is Gδ in 2 ..."
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Cited by 10 (1 self)
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We investigate which definable separable metric spaces are countable dense homogeneous (CDH). We prove that a Borel CDH space is completely metrizable and give a complete list of zerodimensional Borel CDH spaces. We also show that for a Borel X ⊆ 2 ω the following are equivalent: (1) X is Gδ in 2 ω, (2) X ω is CDH and (3) X ω is homeomorphic to 2 ω or to ω ω. Assuming the Axiom of Projective Determinacy the results extend to all projective sets and under the Axiom of Determinacy to all separable metric spaces. In particular, modulo large cardinal assumption it is relatively consistent with ZF that all CDH separable metric spaces are completely metrizable. We also answer a question of Steprāns and Zhou by showing that p = min{κ: 2 κ is not CDH}. A separable topological space X is countable dense homogeneous (CDH) if given any two countable dense subsets D, D ′ ⊆ X there is a homeomorphism h of X such that h[D] = D ′. The first result in this area is due to Cantor, who, in effect, showed that the reals are CDH. Fréchet [Fr] and Brower [Br], independently, proved
Homogeneous spaces and transitive actions by Polish groups
 ISRAEL J. MATH
, 2008
"... We prove that for every homogeneous and strongly locally homogeneous Polish space X there is a Polish group admitting a transitive action on X. We also construct an example of a homogeneous Polish space which is not a coset space and on which no separable metrizable topological group acts transitive ..."
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Cited by 9 (5 self)
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We prove that for every homogeneous and strongly locally homogeneous Polish space X there is a Polish group admitting a transitive action on X. We also construct an example of a homogeneous Polish space which is not a coset space and on which no separable metrizable topological group acts transitively.
Approximation results for reflectionless Jacobi matrices
"... Abstract. We study spaces of reflectionless Jacobi matrices. The main theme is the following type of question: Given a reflectionless Jacobi matrix, is it possible to approximate it by other reflectionless and, typically, simpler Jacobi matrices of a special type? For example, can we approximate by ..."
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Cited by 8 (7 self)
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Abstract. We study spaces of reflectionless Jacobi matrices. The main theme is the following type of question: Given a reflectionless Jacobi matrix, is it possible to approximate it by other reflectionless and, typically, simpler Jacobi matrices of a special type? For example, can we approximate by periodic operators? 1.
Trichotomies for ideals of compact sets
 J. SYMBOLIC LOGIC
"... We prove several trichotomy results for ideals of compact sets. Typically, we show that a “sufficiently rich” universally Baire ideal is either Π 0 3hard, or Σ 0 3hard, or else a σideal. ..."
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Cited by 5 (3 self)
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We prove several trichotomy results for ideals of compact sets. Typically, we show that a “sufficiently rich” universally Baire ideal is either Π 0 3hard, or Σ 0 3hard, or else a σideal.
A note on Ford’s example
 Topology Proc
"... Abstract. Ford gave an example of a homogeneous space that is not a coset space. This example is not metrizable. We present a separable metrizable space with similar properties. 1. ..."
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Cited by 4 (4 self)
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Abstract. Ford gave an example of a homogeneous space that is not a coset space. This example is not metrizable. We present a separable metrizable space with similar properties. 1.
ON THE CHARACTER AND piWEIGHT OF HOMOGENEOUS COMPACTA
, 2003
"... Under GCH, k(X) _< ~(X) for every homogeneous compactum X. CH implies that a homogeneous compactum of countable ~rweight is first countable. There is a compact space of countable 7rweight and uncountable character which is homogeneous nder MA+~CH, but not under CH. ..."
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Cited by 4 (0 self)
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Under GCH, k(X) _< ~(X) for every homogeneous compactum X. CH implies that a homogeneous compactum of countable ~rweight is first countable. There is a compact space of countable 7rweight and uncountable character which is homogeneous nder MA+~CH, but not under CH.
Analytic groups and pushing small sets apart
 Trans. Amer. Math. Soc
"... Abstract. We say that a space X has the separation property provided that if A and B are subsets of X with A countable and B first category, then there is a homeomorphism f: X → X such that f(A) ∩ B = ∅. We prove that a Borel space with this property is Polish. Our main result is that if the homeom ..."
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Cited by 3 (0 self)
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Abstract. We say that a space X has the separation property provided that if A and B are subsets of X with A countable and B first category, then there is a homeomorphism f: X → X such that f(A) ∩ B = ∅. We prove that a Borel space with this property is Polish. Our main result is that if the homeomorphisms needed in the separation property for the space X come from the homeomorphisms given by an action of an analytic group, then X is Polish. Several examples are also presented. 1.
Countable dense homogeneity of definable spaces
 Proc. Amer. Math. Soc
"... Abstract. We investigate which definable separable metric spaces are countable dense homogeneous (CDH). We prove that a Borel CDH space is completely metrizable and give a complete list of zerodimensional Borel CDH spaces. We also show that for a Borel X ⊆ 2 ω the following are equivalent: (1) X is ..."
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Cited by 3 (0 self)
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Abstract. We investigate which definable separable metric spaces are countable dense homogeneous (CDH). We prove that a Borel CDH space is completely metrizable and give a complete list of zerodimensional Borel CDH spaces. We also show that for a Borel X ⊆ 2 ω the following are equivalent: (1) X is Gδ in 2 ω, (2) X ω is CDH and (3) X ω is homeomorphic to 2 ω or to ω ω. Assuming the Axiom of Projective Determinacy the results extend to all projective sets and under the Axiom of Determinacy to all separable metric spaces. In particular, modulo large cardinal assumption it is relatively consistent with ZF that all CDH separable metric spaces are completely metrizable. We also answer a question of Steprāns and Zhou by showing that p = min{κ: 2 κ is not CDH}. A separable topological space X is countable dense homogeneous (CDH) if given any two countable dense subsets D, D ′ ⊆ X there is a homeomorphism h of X such that h[D] = D ′. The first result in this area is due to Cantor, who, in effect, showed that the reals are CDH. Fréchet [Fr] and Brower [Br], independently, proved