Showing papers in "Israel Journal of Mathematics in 2008"

Dehn filling in relatively hyperbolic groups

Abstract: We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we construct two useful bicombings on this space. The first of these, preferred paths, is combinatorial in nature and allows us to define the second, a relatively hyperbolic version of a construction of Mineyev. As an application, we prove a group-theoretic analog of the Gromov-Thurston 2π Theorem in the context of relatively hyperbolic groups.

Classification results for biharmonic submanifolds in spheres

Abstract: We study biharmonic submanifolds of the Euclidean sphere that satisfy certain geometric properties. We classify: (i) the biharmonic hypersurfaces with at most two distinct principal curvatures; (ii) the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudoumbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curvature vector field. We also study the type, in the sense of B-Y. Chen, of compact proper biharmonic submanifolds with constant mean curvature in spheres.

A homotopy theory for stacks

Abstract: We give a homotopy theoretic characterization of stacks on a site C as the homotopy sheaves of groupoids on C. We use this characterization to construct a model category in which stacks are the fibrant objects. We compare different definitions of stacks and show that they lead to Quillen equivalent model categories. In addition, we show that these model structures are Quillen equivalent to the S2-nullification of Jardine’s model structure on sheaves of simplicial sets on C.

On two Hamilton cycle problems in random graphs

Abstract: We study two problems related to the existence of Hamilton cycles in random graphs. The first question relates to the number of edge disjoint Hamilton cycles that the random graph G n,p contains. δ(G)/2 is an upper bound and we show that if p ≤ (1 + o(1)) ln n/n then this upper bound is tight whp. The second question relates to how many edges can be adversarially removed from G n,p without destroying Hamiltonicity. We show that if p ≥ K ln n/n then there exists a constant α > 0 such that whp G − H is Hamiltonian for all choices of H as an n-vertex graph with maximum degree Δ(H) ≤ αK ln n.

Powers of rationals modulo 1 and rational base number systems

Abstract: A new method for representing positive integers and real numbers in a rational base is considered. It amounts to computing the digits from right to left, least significant first. Every integer has a unique expansion. The set of expansions of the integers is not a regular language but nevertheless addition can be performed by a letter-to-letter finite right transducer. Every real number has at least one such expansion and a countable infinite number of them have more than one. We explain how these expansions can be approximated and characterize the expansions of reals that have two expansions. The results that we derive are pertinent on their own and also as they relate to other problems in combinatorics and number theory. A first example is a new interpretation and expansion of the constant K(p) from the so-called “Josephus problem.” More important, these expansions in the base p a llow us to make some progress in the problem of the distribution of the fractional part of the powers of rational numbers.

Noncommutative Burkholder/Rosenthal inequalities II: Applications

Abstract: We show norm estimates for the sum of independent random variables in noncommutative L p -spaces for 1 < p < ∞, following our previous work. These estimates generalize the classical Rosenthal inequality in the commutative case. As applications, we derive an equivalence for the p-norm of the singular values of a random matrix with independent entries, and characterize those symmetric subspaces and unitary ideals which can be realized as subspaces of a noncommutative L p for 2 < p < ∞.

Face vectors of flag complexes

Abstract: A conjecture of Kalai and Eckhoff that the face vector of an arbitrary flag complex is also the face vector of some particular balanced complex is verified.

Locally finite Leavitt path algebras

Abstract: A group-graded K-algebra A = ⊕ g∈G A g is called locally finite in case each graded component A g is finite dimensional over K. We characterize the graphs E for which the Leavitt path algebra L K (E) is locally finite in the standard ℤ-grading. For a locally finite ℤ-graded algebra A we show that, if every nonzero graded ideal has finite codimension in A, then every nonzero ideal has finite codimension in A; that is, ℤ-graded just infinite implies just infinite. We use this result to characterize the finite graphs E for which the Leavitt path algebra L K (E) is locally finite just infinite. We then give an explicit description of the graphs and algebras which arise in this way. In particular, we show that the locally finite Leavitt path algebras are precisely the noetherian Leavitt path algebras.

On metrizable enveloping semigroups

Abstract: When a topological group G acts on a compact space X, its enveloping semigroup E(X) is the closure of the set of g-translations, g ∈ G, in the compact space X X . Assume that X is metrizable. It has recently been shown by the first two authors that the following conditions are equivalent: (1) X is hereditarily almost equicontinuous; (2) X is hereditarily nonsensitive; (3) for any compatible metric d on X the metric d G (x, y) ≔ sup{d(gx, gy): g ∈ G} defines a separable topology on X; (4) the dynamical system (G, X) admits a proper representation on an Asplund Banach space. We prove that these conditions are also equivalent to the following: the enveloping semigroup E(X) is metrizable.

Weights in serre's conjecture for hilbert modular forms: the ramified case

Abstract: Let F be a totally real field and p ≥ 3 a prime. If ρ : Open image in new window is continuous, semisimple, totally odd, and tamely ramified at all places of F dividing p, then we formulate a conjecture specifying the weights for which ρ is modular. This extends the conjecture of Diamond, Buzzard, and Jarvis, which required p to be unramified in F. We also prove a theorem that verifies one half of the conjecture in many cases and use Dembele’s computations of Hilbert modular forms over \(\mathbb{Q}(\sqrt 5 )\) to provide evidence in support of the conjecture.

The ratio set of the harmonic measure of a random walk on a hyperbolic group

Abstract: We consider the harmonic measure on the Gromov boundary of a non-amenable hyperbolic group defined by a finite range random walk on the group, and study the corresponding orbit equivalence relation on the boundary. It is known to be always amenable and of type III. We determine its ratio set by showing that it is generated by certain values of the Martin kernel. In particular, we show that the equivalence relation is never of type III0.

Toward the theory of ring q-homeomophisms

Abstract: We investigate classes of the so-called ring Q-homeomorphisms including, in particular, Q-homeomorphisms, various classes of homeomorphisms with finite length distortion, Sobolev’s classes etc. In terms of the majorant Q(x), we give a series of criteria for normality based on estimates of the distortion of the spherical distance under ring Q-homeomorphisms. In particular, it is shown that the class $$ \Re _{Q,\Delta } $$ of all ring Q-homeomorphisms f of a domain D ⊂ ℝ n into , n ≥ 2, with , forms a normal family, if Q(x) has finite mean oscillation in D. We also prove normality of $$ \Re _{Q,\Delta } $$ , for instance, if Q(x) has singularities of logarithmic type whose degrees are not greater than n − 1 at every point x ∈ D. The results are applicable, in particular, to mappings with finite length distortion and Sobolev’s classes.

Reducibility of rational functions in several variables

Abstract: We prove an analogous of Stein theorem for rational functions in several variables: we bound the number of reducible fibers by a formula depending on the degree of the fraction.

Generalized Humbert curves

Abstract: In this note we consider a certain class of closed Riemann surfaces which are a natural generalization of the so called classical Humbert curves. They are given by closed Riemann surfaces S admitting H ≅ ℤ 2 k as a group of conformal automorphisms so that S/H is an orbifold of signature (0, k + 1; 2,…, 2). The classical ones are given by k = 4. Mainly, we describe some of its generalities and provide Fuchsian, algebraic and Schottky descriptions.

On distance sets of large sets of integer points

Abstract: Distance sets of large sets of integer points are studied in dimensions at least 5. To any e > 0 a positive integer Q ɛ is constructed with the following property; If A is any set of integer points of upper density at least e, then all large multiples of Q ɛ 2 occur as squares of distances between the points of the set A.

Rotational linear Weingarten surfaces of hyperbolic type

Abstract: A linear Weingarten surface in Euclidean space R 3 is a surface whose mean curvature H and Gaussian curvature K satisfy a relation of the form aH + bK = c, where a, b, c 2 R. Such a surface is said to be hyperbolic when a 2 + 4bc < 0. In this paper we study rotational linear Weingarten surfaces of hyperbolic type giving a classification under suitable hypothesis. As a consequence, we obtain a family of complete hyperbolic linear Weingarten surfaces in R 3 that consists of surfaces with self-intersections whose generating curves are periodic.

A proof of Yomdin-Gromov’s Algebraic Lemma

Abstract: Following the analysis of differentiable mappings of Y. Yomdin, M. Gromov has stated a very elegant “Algebraic Lemma” which says that the “differentiable size” of an algebraic subset may be bounded only in terms of its dimension, degree and diameter, regardless of the size and specific values of the underlying coefficients. We give a complete and elementary proof of Gromov’s result.

On the structure of theta lifts of discrete series for dual pairs (Sp(n), O(V))

Abstract: The purpose of this paper is to prove some fundamental results on the structure of theta lifts of discrete series.

The diffie-hellman key exchange protocol and non-abelian nilpotent groups

Abstract: In this paper we study a key exchange protocol similar to the Diffie-Hellman key exchange protocol, using abelian subgroups of the automorphism group of a non-abelian nilpotent group. We also generalize group no. 92 of the Hall-Senior table [16] to an arbitrary prime p and show that, for those groups, the group of central automorphisms is commutative. We use these for the key exchange we are studying.

OMEGA SUBGROUPS OF PRO-p GROUPS

Abstract: Let G be a pro-p group and let k ≥ 1. If γk(p−1) (G) ≤ γr\((G)^{p^s } \) for some r and s such that k(p − 1) < r + s(p − 1), we prove that the exponent of Ωi(G) is at most pi+k−1 for all i.

Tannaka-Krein duality for Hopf algebroids

Abstract: We show that a Hopf algebroid can be reconstructed from a monoidal functor from a monoidal category into the category of rigid bimodules over a ring. We study the equivalence between the original category and the category of comodules over the reconstructed Hopf algebroid.

Upper bound for topological entropy of a meromorphic correspondence

Abstract: Let f be a meromorphic correspondence on a compact Kahler manifold. We show that the topological entropy of f is bounded from above by the logarithm of its maximal dynamical degree. An analogous estimate for the entropy on subvarieties is given. We also discuss a notion of Julia and Fatou sets.

ON DEFORMATIONS OF Fn IN COMPACT LIE GROUPS

Abstract: We study some properties of the varieties of deformations of free groups in compact Lie groups. In particular, we prove a conjecture of Margulis and Soifer about the density of non-virtually free points in such variety, and a conjecture of Goldman on the ergodicity of the action of Aut(F n ) on such variety when n ≥ 3.

Contravariant functors on the category of finitely presented modules

Abstract: If R is an associative ring with identity, a theory of minimal flat resolutions is developed in the category ((R-mod)op, Ab) of contravariant functors G: (R-mod)op → Ab from the category R-mod of finitely presented left R-modules to the category Ab of abelian groups. For a left R-module M, it is shown that the flat contravariant functor (−, M) is cotorsion if and only if M is pure-injective. This is applied to characterize when a flat resolution of an object F in ((R-mod)op, Ab) is minimal, and is used to construct a minimal flat resolution of F, given a projective presentation.

Jacobians with a vanishing theta-null in genus 4

Abstract: In this paper we prove a conjecture of Hershel Farkas [11] that if a 4-dimensional principally polarized abelian variety has a vanishing theta-null, and the Hessian of the theta function at the corresponding 2-torsion point is degenerate, the abelian variety is a Jacobian.

Strong shift equivalence of C*-correspondences

Abstract: We define a notion of strong shift equivalence for C*-correspondences and show that strong shift equivalent C*-correspondences have strongly Morita equivalent Cuntz-Pimsner algebras. Our analysis extends the fact that strong shift equivalent square matrices with non-negative integer entries give stably isomorphic Cuntz-Krieger algebras.

Reality properties of conjugacy classes in algebraic groups

Abstract: Let G be an algebraic group defined over a field k. We call g ∈ G real if g is conjugate to g −1 and g ∈ G(k) as k-real if g is real in G(k). An element g ∈ G is strongly real if ∃h ∈ G, h 2 = 1 (i.e., h is an involution) such that hgh −1 = g −1. Clearly, strongly real elements are real and are product of two involutions. Let G be a connected adjoint semisimple group over a perfect field k, with −1 in the Weyl group. We prove that any strongly regular k-real element in G(k) is strongly k-real (i.e., is a product of two involutions in G(k)). For classical groups, with some mild exceptions, over an arbitrary field k of characteristic not 2, we prove that k-real semisimple elements are strongly k-real. We compute an obstruction to reality and prove some results on reality specific to fields k with cd(k) ≤ 1. Finally, we prove that in a group G of type G 2 over k, characteristic of k different from 2 and 3, any real element in G(k) is strongly k-real. This extends our results in [ST05], on reality for semisimple and unipotent real elements in groups of type G 2.

Optimisers for the Brascamp-Lieb inequality

Abstract: We find all optimisers for the Brascamp-Lieb inequality, thus completing the problem which was settled in special cases by Barthe; Carlen, Lieb and Loss; and Bennett, Carbery, Christ and Tao. Our approach to the solution is based on the heat flow methods introduced by the second and third sets of authors above. We present the heat flow method in the form which is most appropriate for our study and also expand the structural theory for the Brascamp-Lieb inequality as necessary for the description of the optimisers.

The automorphism group of certain factorial threefolds and a cancellation problem

Abstract: A new class of counterexamples to a generalized cancellation problem for affine varieties is presented. Each member of the class is an affine factorial complex threefold admitting a locally trivial action of the additive group, hence the total space for a principal G a bundle over a quasiaffine base. The automorphism groups for these varieties are also determined.

Automatic continuity in homeomorphism groups of compact 2-manifolds

Abstract: We show that any homomorphism from the homeomorphism group of a compact 2-manifold, with the compact-open topology, or equivalently, with the topology of uniform convergence, into a separable topological group is automatically continuous.