
20The Nonexistence of a Binary Homogeneous PseudoplaneMathematical Logic Quarterly 44 (1): 135137. 1998.We prove that there are no binary homogeneous pseudoplanes

17The biembeddability relation for finitely generated groups IIArchive for Mathematical Logic 55 (34): 385396. 2016.We study the isomorphism and biembeddability relations on the spaces of Kazhdan groups and finitely generated simple groups.

27Superrigidity and countable Borel equivalence relationsAnnals of Pure and Applied Logic 120 (13): 237262. 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 padic fields are pairwise incomparable with respect to Borel reducibility

37Reducts of random hypergraphsAnnals of Pure and Applied Logic 80 (2): 165193. 1996.For each k 1, let Γk be the countable universal homogeneous khypergraph. 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

43Martin’s conjecture and strong ergodicityArchive for Mathematical Logic 48 (8): 749759. 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

20Groupwise density and the cofinality of the infinite symmetric groupArchive for Mathematical Logic 37 (7): 483493. 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 $c(Sym(\omega)) = \omega_{1}$

42Continuous versus Borel reductionsArchive for Mathematical Logic 48 (8): 761770. 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

43A descriptive view of combinatorial group theoryBulletin of Symbolic Logic 17 (2): 252264. 2011.In this paper, we will prove the inevitable nonuniformity of two constructions from combinatorial group theory related to the word problem for finitely generated groups and the Higman—Neumann—Neumann Embedding Theorem

33Unbounded families and the cofinality of the infinite symmetric groupArchive for Mathematical Logic 34 (1): 3345. 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

42Changing the heights of automorphism towersAnnals of Pure and Applied Logic 102 (12): 139157. 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

17The classification problem for plocal torsionfree Abelian groups of rank twoJournal of Mathematical Logic 6 (2): 233251. 2006.We prove that if p ≠ q are distinct primes, then the classification problems for plocal and qlocal torsionfree abelian groups of rank two are incomparable with respect to Borel reducibility.

20Complete groups are complete coanalyticArchive for Mathematical Logic 57 (56): 601606. 2018.The set of complete groups is a complete coanalytic subset of the standard Borel space of countably infinite groups.

10The cofinality spectrum of the infinite symmetric groupJournal of Symbolic Logic 62 (3): 902916. 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

212002 european summer meeting of the association for symbolic logic logic colloquium'02Bulletin of Symbolic Logic 9 (1): 71. 2003.

20Popa superrigidity and countable Borel equivalence relationsAnnals of Pure and Applied Logic 158 (3): 175189. 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

24San Diego Convention Center, San Diego, CA January 8–9, 2008Bulletin of Symbolic Logic 14 (3). 2008.

26University of California at Berkeley Berkeley, CA, USA March 24–27, 2011Bulletin of Symbolic Logic 18 (2). 2012.