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Simon Thomas
Queen Mary and Westfield College, University of London
  1.  15
    Superrigidity and Countable Borel Equivalence Relations.Simon Thomas - 2003 - Annals of Pure and Applied Logic 120 (1-3):237-262.
    We formulate a Borel version of a corollary of Furman's superrigidity theorem for orbit equivalence and present a number of applications to the theory of countable Borel equivalence relations. In particular, we prove that the orbit equivalence relations arising from the natural actions of on the projective planes over the various p-adic fields are pairwise incomparable with respect to Borel reducibility.
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  2.  28
    Martin’s Conjecture and Strong Ergodicity.Simon Thomas - 2009 - Archive for Mathematical Logic 48 (8):749-759.
    In this paper, we explore some of the consequences of Martin’s Conjecture on degree invariant Borel maps. These include the strongest conceivable ergodicity result for the Turing equivalence relation with respect to the filter on the degrees generated by the cones, as well as the statement that the complexity of a weakly universal countable Borel equivalence relation always concentrates on a null set.
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  3.  8
    Groupwise Density and the Cofinality of the Infinite Symmetric Group.Simon Thomas - 1998 - Archive for Mathematical Logic 37 (7):483-493.
    We study the relationship between the cofinality $c(Sym(\omega))$ of the infinite symmetric group and the cardinal invariants $\frak{u}$ and $\frak{g}$ . In particular, we prove the following two results. Theorem 0.1 It is consistent with ZFC that there exists a simple $P_{\omega_{1}}$ -point and that $c(Sym(\omega)) = \omega_{2} = 2^{\omega}$ . Theorem 0.2 If there exist both a simple $P_{\omega_{1}}$ -point and a $P_{\omega_{2}}$ -point, then $c(Sym(\omega)) = \omega_{1}$.
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  4. On the Complexity of the Classification Problem for Torsion-Free Abelian Groups of Finite Rank.Simon Thomas - 2001 - Bulletin of Symbolic Logic 7 (3):329-344.
  5.  15
    Reducts of Random Hypergraphs.Simon Thomas - 1996 - Annals of Pure and Applied Logic 80 (2):165-193.
    For each k 1, let Γk be the countable universal homogeneous k-hypergraph. In this paper, we shall classify the closed permutation groups G such that Aut G Sym. In particular, we shall show that there exist only finitely many such groups G for each k 1. We shall also show that each of the associated reducts of Γk is homogeneous with respect to a finite relational language.
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  6.  9
    Popa Superrigidity and Countable Borel Equivalence Relations.Simon Thomas - 2009 - Annals of Pure and Applied Logic 158 (3):175-189.
    We present some applications of Popa’s Superrigidity Theorem to the theory of countable Borel equivalence relations. In particular, we show that the universal countable Borel equivalence relation E∞ is not essentially free.
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  7.  23
    Reducts of the Random Graph.Simon Thomas - 1991 - Journal of Symbolic Logic 56 (1):176-181.
  8. The Cofinality Spectrum of the Infinite Symmetric Group.Saharon Shelah & Simon Thomas - 1997 - Journal of Symbolic Logic 62 (3):902-916.
    Let S be the group of all permutations of the set of natural numbers. The cofinality spectrum CF(S) of S is the set of all regular cardinals λ such that S can be expressed as the union of a chain of λ proper subgroups. This paper investigates which sets C of regular uncountable cardinals can be the cofinality spectrum of S. The following theorem is the main result of this paper. Theorem. Suppose that $V \models GCH$ . Let C be (...)
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  9.  8
    The Bi-Embeddability Relation for Finitely Generated Groups II.Simon Thomas & Jay Williams - 2016 - Archive for Mathematical Logic 55 (3-4):385-396.
    We study the isomorphism and bi-embeddability relations on the spaces of Kazhdan groups and finitely generated simple groups.
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  10.  8
    Unbounded Families and the Cofinality of the Infinite Symmetric Group.James D. Sharp & Simon Thomas - 1995 - Archive for Mathematical Logic 34 (1):33-45.
    In this paper, we study the relationship between the cofinalityc(Sym(ω)) of the infinite symmetric group and the minimal cardinality $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{b} $$ of an unbounded familyF of ω ω.
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  11.  5
    Uniformization Problems and the Cofinality of the Infinite Symmetric Group.James D. Sharp & Simon Thomas - 1994 - Notre Dame Journal of Formal Logic 35 (3):328-345.
    Assuming Martin's Axiom, we compute the value of the cofinality of the symmetric group on the natural numbers. We also show that Martin's Axiom does not decide the value of the covering number of a related Mycielski ideal.
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  12.  30
    Changing the Heights of Automorphism Towers.Joel David Hamkins & Simon Thomas - 2000 - Annals of Pure and Applied Logic 102 (1-2):139-157.
    If G is a centreless group, then τ denotes the height of the automorphism tower of G. We prove that it is consistent that for every cardinal λ and every ordinal α<λ, there exists a centreless group G such that τ=α; and if β is any ordinal such that 1β<λ, then there exists a notion of forcing , which preserves cofinalities and cardinalities, such that τ=β in the corresponding generic extension.
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  13.  9
    The Classification Problem for P-Local Torsion-Free Abelian Groups of Rank Two.Greg Hjorth & Simon Thomas - 2006 - Journal of Mathematical Logic 6 (2):233-251.
    We prove that if p ≠ q are distinct primes, then the classification problems for p-local and q-local torsion-free abelian groups of rank two are incomparable with respect to Borel reducibility.
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  14.  33
    Continuous Versus Borel Reductions.Simon Thomas - 2009 - Archive for Mathematical Logic 48 (8):761-770.
    We present some natural examples of countable Borel equivalence relations E, F with E ≤ B F such that there does not exist a continuous reduction from E to F.
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  15.  24
    Theories with Finitely Many Models.Simon Thomas - 1986 - Journal of Symbolic Logic 51 (2):374-376.
  16.  27
    Subgroups of Small Index in Infinite Symmetric Groups. II.Saharon Shelah & Simon Thomas - 1989 - Journal of Symbolic Logic 54 (1):95-99.
  17.  25
    Homogeneity of Infinite Permutation Groups.Saharon Shelah & Simon Thomas - 1989 - Archive for Mathematical Logic 28 (2):143-147.
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  18.  18
    Maximal Subgroups of Infinite Symmetric Groups.James E. Baumgartner, Saharon Shelah & Simon Thomas - 1992 - Notre Dame Journal of Formal Logic 34 (1):1-11.
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  19.  12
    Complete Groups Are Complete Co-Analytic.Simon Thomas - 2018 - Archive for Mathematical Logic 57 (5-6):601-606.
    The set of complete groups is a complete co-analytic subset of the standard Borel space of countably infinite groups.
    No categories
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  20.  16
    San Diego Convention Center, San Diego, CA January 8–9, 2008.Gregory L. Cherlin, Ilijas Farah, Pavel Hrubes, Victor Marek, Jan Riemann, Simon Thomas & Jeffrey Remmel - 2008 - Bulletin of Symbolic Logic 14 (3).
  21.  10
    2002 European Summer Meeting of the Association for Symbolic Logic Logic Colloquium'02.Lev D. Beklemishev, Stephen Cook, Olivier Lessmann, Simon Thomas, Jeremy Avigad, Arnold Beckmann, Tim Carlson, Robert L. Constable & Kosta Došen - 2003 - Bulletin of Symbolic Logic 9 (1):71.
  22.  16
    A Descriptive View of Combinatorial Group Theory.Simon Thomas - 2011 - Bulletin of Symbolic Logic 17 (2):252-264.
    In this paper, we will prove the inevitable non-uniformity of two constructions from combinatorial group theory related to the word problem for finitely generated groups and the Higman—Neumann—Neumann Embedding Theorem.
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  23.  10
    University of California at Berkeley Berkeley, CA, USA March 24–27, 2011.G. Aldo Antonelli, Laurent Bienvenu, Lou van den Dries, Deirdre Haskell, Justin Moore, Christian Rosendal Uic, Neil Thapen & Simon Thomas - 2012 - Bulletin of Symbolic Logic 18 (2).
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  24.  8
    University of California, San Diego, March 20–23, 1999.Julia F. Knight, Steffen Lempp, Toniann Pitassi, Hans Schoutens, Simon Thomas, Victor Vianu & Jindrich Zapletal - 1999 - Bulletin of Symbolic Logic 5 (3).
  25.  14
    Some Questions Concerning the Confinality of Sym (K).James D. Sharp & Simon Thomas - 1995 - Journal of Symbolic Logic 60 (3):892-897.
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  26.  12
    Property Τ and Countable Borel Equivalence Relations.Simon Thomas - 2007 - Journal of Mathematical Logic 7 (1):1-34.
    We prove Borel superrigidity results for suitably chosen actions of groups of the form SL2, 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 on the projective lines ℚp ∪ {∞}, p ≠ q, over the various p-adic fields are pairwise incomparable with respect to (...)
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  27.  11
    Two Cardinal Properties of Homogeneous Graphs.Gregory Cherlin & Simon Thomas - 2002 - Journal of Symbolic Logic 67 (1):217-220.
    We analyze the two cardinal properties of definable sets in homogeneous graphs.
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  28.  6
    The Nonexistence of a Binary Homogeneous Pseudoplane.Simon Thomas - 1998 - Mathematical Logic Quarterly 44 (1):135-137.
    We prove that there are no binary homogeneous pseudoplanes.
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