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39Changing the heights of automorphism towersAnnals of Pure and Applied Logic 102 (1-2): 139-157. 2000.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
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18Complete groups are complete co-analyticArchive for Mathematical Logic 57 (5-6): 601-606. 2018.The set of complete groups is a complete co-analytic subset of the standard Borel space of countably infinite groups.
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29Maximal subgroups of infinite symmetric groupsNotre Dame Journal of Formal Logic 34 (1): 1-11. 1992.
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13The Nonexistence of a Binary Homogeneous PseudoplaneMathematical Logic Quarterly 44 (1): 135-137. 1998.We prove that there are no binary homogeneous pseudoplanes
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38Continuous versus Borel reductionsArchive for Mathematical Logic 48 (8): 761-770. 2009.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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15Groupwise density and the cofinality of the infinite symmetric groupArchive for Mathematical Logic 37 (7): 483-493. 1998.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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42Martin’s conjecture and strong ergodicityArchive for Mathematical Logic 48 (8): 749-759. 2009.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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38Subgroups of small index in infinite symmetric groups. IIJournal of Symbolic Logic 54 (1): 95-99. 1989.
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32Some questions concerning the confinality of sym (k)Journal of Symbolic Logic 60 (3): 892-897. 1995.
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219The cofinality spectrum of the infinite symmetric groupJournal of Symbolic Logic 62 (3): 902-916. 1997.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 a set of regular uncountable cardinals which satisfies the following …Read more
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31Unbounded families and the cofinality of the infinite symmetric groupArchive for Mathematical Logic 34 (1): 33-45. 1995.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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10The cofinality spectrum of the infinite symmetric groupJournal of Symbolic Logic 62 (3): 902-916. 1997.LetSbe the group of all permutations of the set of natural numbers. The cofinality spectrumCF(S)ofSis the set of all regular cardinalsλsuch thatScan be expressed as the union of a chain ofλproper subgroups. This paper investigates which setsCof regular uncountable cardinals can be the cofinality spectrum ofS. The following theorem.is the main result of this paper.Theorem.Suppose that V ⊨ GCH. Let C be a set of regular uncountable cardinals which satisfies the following conditions.(a)C contains a…Read more
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22Uniformization Problems and the Cofinality of the Infinite Symmetric GroupNotre Dame Journal of Formal Logic 35 (3): 328-345. 1994.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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21Two cardinal properties of homogeneous graphsJournal of Symbolic Logic 67 (1): 217-220. 2002.We analyze the two cardinal properties of definable sets in homogeneous graphs
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36Reducts of random hypergraphsAnnals of Pure and Applied Logic 80 (2): 165-193. 1996.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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24San Diego Convention Center, San Diego, CA January 8–9, 2008Bulletin of Symbolic Logic 14 (3). 2008.
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38A descriptive view of combinatorial group theoryBulletin of Symbolic Logic 17 (2): 252-264. 2011.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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22Superrigidity and countable Borel equivalence relationsAnnals of Pure and Applied Logic 120 (1-3): 237-262. 2003.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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13The classification problem for p-local torsion-free Abelian groups of rank twoJournal of Mathematical Logic 6 (2): 233-251. 2006.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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161On the complexity of the classification problem for torsion-free Abelian groups of finite rankBulletin of Symbolic Logic 7 (3): 329-344. 2001.In this paper, we shall discuss some recent contributions to the project [15, 14, 2, 18, 22, 23] of explaining why no satisfactory system of complete invariants has yet been found for the torsion-free abelian groups of finite rank n ≥ 2. Recall that, up to isomorphism, the torsion-free abelian groups of rank n are exactly the additive subgroups of the n-dimensional vector space ℚn which contain n linearly independent elements. Thus the collection of torsion-free abelian groups of rank at most n …Read more
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26University of California at Berkeley Berkeley, CA, USA March 24–27, 2011Bulletin of Symbolic Logic 18 (2). 2012.
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15The bi-embeddability relation for finitely generated groups IIArchive for Mathematical Logic 55 (3-4): 385-396. 2016.We study the isomorphism and bi-embeddability relations on the spaces of Kazhdan groups and finitely generated simple groups.
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18Popa superrigidity and countable Borel equivalence relationsAnnals of Pure and Applied Logic 158 (3): 175-189. 2009.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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212002 european summer meeting of the association for symbolic logic logic colloquium'02Bulletin of Symbolic Logic 9 (1): 71. 2003.
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17Property τ and countable borel equivalence relationsJournal of Mathematical Logic 7 (1): 1-34. 2007.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 Borel reducibility.