9 found
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  1.  40
    Cofinal families of Borel equivalence relations and quasiorders.Christian Rosendal - 2005 - Journal of Symbolic Logic 70 (4):1325-1340.
    Families 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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  2. Automatic continuity of group homomorphisms.Christian Rosendal - 2009 - Bulletin of Symbolic Logic 15 (2):184-214.
    We survey various aspects of the problem of automatic continuity of homomorphisms between Polish groups.
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  3.  13
    Compact Metrizable Structures and Classification Problems.Christian Rosendal & Joseph Zielinski - 2018 - Journal of Symbolic Logic 83 (1):165-186.
    We introduce and study the framework of compact metric structures and their associated notions of isomorphisms such as homeomorphic and bi-Lipschitz isomorphism. This is subsequently applied to model various classification problems in analysis such as isomorphism ofC*-algebras and affine homeomorphism of Choquet simplices, where among other things we provide a simple proof of the completeness of the isomorphism relation of separable, simple, nuclearC*-algebras recently established by M. Sabok.
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  4.  41
    Finitely approximable groups and actions Part II: Generic representations.Christian Rosendal - 2011 - Journal of Symbolic Logic 76 (4):1307-1321.
    Given a finitely generated group Γ, we study the space Isom(Γ, ℚ������) of all actions of Γ by isometries of the rational Urysohn metric space ℚ������, where Isom(Γ, ℚ������) is equipped with the topology it inherits seen as a closed subset of Isom(ℚ������) Γ . When Γ is the free group ������ n on n generators this space is just Isom(ℚ������) n , but is in general significantly more complicated. We prove that when Γ is finitely generated Abelian there is (...)
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  5.  49
    Finitely approximable groups and actions Part I: The Ribes—Zaluesskiĭ property.Christian Rosendal - 2011 - Journal of Symbolic Logic 76 (4):1297-1306.
    We investigate extensions of S. Solecki's theorem on closing off finite partial isometries of metric spaces [11] and obtain the following exact equivalence: any action of a discrete group Γ by isometries of a metric space is finitely approximable if and only if any product of finitely generated subgroups of Γ is closed in the profinite topology on Γ.
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  6.  70
    The complexity of continuous embeddability between dendrites.Alberto Marcone & Christian Rosendal - 2004 - Journal of Symbolic Logic 69 (3):663-673.
    We show that the quasi-order of continuous embeddability between finitely branching dendrites (a natural class of fairly simple compacta) is $\Sigma_1^1$ -complete. We also show that embeddability between countable linear orders with infinitely many colors is $\Sigma_1^1$ -complete.
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  7.  34
    Completely metrisable groups acting on trees.Christian Rosendal - 2011 - Journal of Symbolic Logic 76 (3):1005 - 1022.
    We consider actions of completely metrisable groups on simplicial trees in the context of the Bass—Serre theory. Our main result characterises continuity of the amplitude function corresponding to a given action. Under fairly mild conditions on a completely metrisable group G, namely, that the set of elements generating a non-discrete or finite subgroup is somewhere dense, we show that in any decomposition as a free product with amalgamation, G = A * C B, the amalgamated groups A, B and C (...)
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  8.  15
    Superstability and Categoricity in Abstract Elementary Classes, Carnegie Mellon University, USA, 2017. Supervised by Rami Grossberg.Christian Rosendal & Sebastien Vasey - 2018 - Bulletin of Symbolic Logic 24 (2):192-194.
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  9.  36
    Universally measurable subgroups of countable index.Christian Rosendal - 2010 - Journal of Symbolic Logic 75 (3):1081-1086.
    It is proved that any countable index, universally measurable subgroup of a Polish group is open. By consequence, any universally measurable homomorphism from a Polish group into the infinite symmetric group S ∞ is continuous. It is also shown that a universally measurable homomorphism from a Polish group into a second countable, locally compact group is necessarily continuous.
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