Kazhdan Constants of Hyperbolic Groups

Let H be an infinite hyperbolic group with Kazhdan property (T) and let (H,X) denote the Kazhdan constant of H with respect to a generating set X. We prove that inf_X(H,X)=0, where the infimum is taken over all finite generating sets of H. In particular, this gives an answer to a Lubotzky question.     

Weakly Hyperbolic Actions of Kazhdan Groups on Tori

We study the ergodic and rigidity properties of weakly hyperbolic actions. First, we establish ergodicity for C² volume preserving weakly hyperbolic group actions on closed manifolds. For the integral action generated by a single Anosov diffeomorphism this theorem is classical and originally due to Anosov. Motivated by the Franks/Manning classification of Anosov diffeomorphisms on tori, we restrict our attention to weakly hyperbolic actions on the torus. When the acting group is a lattice subgroup of a semisimple Lie group with no compact factors and all (almost) simple factors of real rank at least two, we show that weak hyperbolicity in the original action implies weak hyperbolicity for the induced action on the fundamental group. As a corollary, we obtain that any such action on the torus is continuously semiconjugate to the affine action coming from the fundamental group via a map unique in the homotopy class of the identity. Under the additional assumption that some partially hyperbolic group element has quasi-isometrically embedded lifts of unstable leaves to the universal cover, we obtain a conjugacy, resulting in a continuous classification for these actions. Partially funded by VIGRE grant DMS-9977371 Received: January 2005 Revision: August 2005 Accepted: September 2005     

On the finite dimensional unitary representations of Kazhdan groups

We use A. Weil's criterion to prove that all finite dimensional unitary representations of a discrete Kazhdan group are locally rigid. It follows that any such representation is unitarily equivalent to a unitary representation over some algebraic number field.

     

Groups acting on cubes and Kazhdan's property (T)

We show that a group contains a subgroup with if and only if it admits an action on a connected cube that is transitive on the hyperplanes and has no fixed point. As a corollary we deduce that a countable group with such a subgroup does not satisfy Kazhdan's property (T).

     

On the notion of relative property (T) for inclusions of von Neumann algebras

We prove that the notion of rigidity (or relative property (T)) for inclusions of finite von Neumann algebras recently defined by the second author is equivalent to a weaker property, in which no “continuity constants” are required. The proof is by contradiction and uses infinite products of completely positive maps, regarded as correspondences.     

Uniform Kazhdan constant for some families of linear groups

Let R be a ring generated by l elements with stable range r. Assume that the group EL_d(R) has Kazhdan constant ₀>0 for some dr+1. We prove that there exist (₀,l)>0 and , s.t. for every nd, EL_n(R) has a generating set of order k and a Kazhdan constant larger than . As a consequence, we obtain for where n3, a Kazhdan constant which is independent of n w.r.t. generating set of a fixed size.     

The product replacement algorithm and Kazhdan's property (T)

The ``product replacement algorithm' is a commonly used heuristic to generate random group elements in a finite group , by running a random walk on generating -tuples of . While experiments showed outstanding performance, the theoretical explanation remained mysterious. In this paper we propose a new approach to the study of the algorithm, by using Kazhdan's property (T) from representation theory of Lie groups.

     

Actions of Semisimple Lie Groups on Circle Bundles

Suppose G is a connected, simple, real Lie group with -rank(G) 2, M is an ergodic G-space with invariant probability measure , and : G × M Homeo( ) is a Borel cocycle. We use an argument of É. Ghys to show that there is a G-invariant probability measure on the skew product M × , such that the projection of to M is . Furthermore, if (G × M) Diff¹( ), then can be taken to be equivalent to × , where is Lebesgue measure on ; therefore, is cohomologous to a cocycle with values in the isometry group of .     

On B(5, 18) Groups

A group G is said to be a B(n, k) group if for any n-element subset A of G, |A²| ≤k. In this paper, a characterization of B(5, 18) groups is given. It is shown that G is a B(5, 18) group if and only if one of the following statements holds: (1) G is abelian; (2) |G| ≤18; (3) G ? ? a, b | a⁵ = b⁴ = 1, a^b = a^?1 ?.     

Rigidity results for wreath product II1 factors

We consider II₁ factors of the form , where either (i) B is a non-hyperfinite II₁ factor and G is an ICC amenable group or (ii) B is a weakly rigid II₁ factor and G is ICC group and where G acts on by Bernoulli shifts. We prove that isomorphism of two such factors implies cocycle conjugacy of the corresponding Bernoulli shift actions. In particular, the groups acting must be isomorphic. As a consequence, we can distinguish between certain classes of group von Neumann algebras associated to wreath product groups.     

五点八边图的完美T(G)-三元系

设G是Kn的子图.在G的每边外添加一点,将该边扩展为一个3长圈,且所添加的点两两不同,均异于G的诸顶点,这样得到的图形被记为T(G).如果3Kn的边恰好能够分拆成与T(G)同构的一些子图,则称这些子图构成一个n阶的T(G)-三元系.进而,若此分拆的全体内部边又恰构成Kn中全部边的一个分拆,则称这个T(G)-三元系是完美的.对于所有使得完美T(G)-三元系存在的正整数n的集合称为完美T(G)-三元系的存在谱.对于K4的所有子图及K5的7边以下子图G,其完美T(G)-三元系的存在性问题已经在一系列文章中被完全解决.本文将对不含孤立点的全部五点八边图G,确定完美T(G)-三元系的存在谱.     

New higher-order numerical one-step methods based on the Adomian and the modified decomposition methods

We develop new, higher-order numerical one-step methods and apply them to several examples to investigate approximate discrete solutions of nonlinear differential equations. These new algorithms are derived from the Adomian decomposition method (ADM) and the Rach-Adomian-Meyers modified decomposition method (MDM) to present an alternative to such classic schemes as the explicit Runge-Kutta methods for engineering models, which require high accuracy with low computational costs for repetitive simulations in contrast to a one-size-fits-all philosophy. This new approach incorporates the notion of analytic continuation, which extends the region of convergence without resort to other techniques that are also used to accelerate the rate of convergence such as the diagonal Padé approximants or the iterated Shanks transforms. Hence global solutions instead of only local solutions are directly realized albeit in a discretized representation. We observe that one of the difficulties in applying explicit Runge-Kutta one-step methods is that there is no general procedure to generate higher-order numeric methods. It becomes a time-consuming, tedious endeavor to generate higher-order explicit Runge-Kutta formulas, because it is constrained by the traditional Picard formalism as used to represent nonlinear differential equations. The ADM and the MDM rely instead upon Adomian’s representation and the Adomian polynomials to permit a straightforward universal procedure to generate higher-order numeric methods at will such as a 12th-order or 24th-order one-step method, if need be. Another key advantage is that we can easily estimate the maximum step-size prior to computing data sets representing the discretized solution, because we can approximate the radius of convergence from the solution approximants unlike the Runge-Kutta approach with its intrinsic linearization between computed data points. We propose new variable step-size, variable order and variable step-size, variable order algorithms for automatic step-size control to increase the computational efficiency and reduce the computational costs even further for critical engineering models.     

The Phillips properties

A Banach space has the Phillips property if the canonical projection is sequentially weak-norm continuous, and has the weak Phillips property if is sequentially weak-weak continuous. We study both properties in connection with other geometric properties, such as the Dunford-Pettis property, Pelczynski's properties and (V), and the Schur property.

     

Quantifying properties (K) and (μs$\mu ^{s}$)

A Banach space X has property (K), whenever every weak* null sequence in the dual space admits a convex block subsequence ${(f_n)}_{n=1}^{\infty }$ so that $⟨f_n,x_n⟩\to 0$ as $n\to \infty$ for every weakly null sequence ${(x_n)}_{n=1}^{\infty }$ in X; X has property $({\mu }^s)$ if every weak* null sequence in $X^{\ast }$ admits a subsequence so that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. Both property $({\mu }^s)$ and reflexivity (or even the Grothendieck property) imply property (K). In this paper, we propose natural ways for quantifying the aforementioned properties in the spirit of recent results concerning other familiar properties of Banach spaces.     

Strong CHIP, normality, and linear regularity of convex sets

We extend the property (N) introduced by Jameson for closed convex cones to the normal property for a finite collection of convex sets in a Hilbert space. Variations of the normal property, such as the weak normal property and the uniform normal property, are also introduced. A dual form of the normal property is derived. When applied to closed convex cones, the dual normal property is the property (G) introduced by Jameson. Normality of convex sets provides a new perspective on the relationship between the strong conical hull intersection property (strong CHIP) and various regularity properties. In particular, we prove that the weak normal property is a dual characterization of the strong CHIP, and the uniform normal property is a characterization of the linear regularity. Moreover, the linear regularity is equivalent to the fact that the normality constant for feasible direction cones of the convex sets at is bounded away from 0 uniformly over all points in the intersection of these convex sets.

     

The single-valued extension property for sums and products of commuting operators

It is shown that the sum and the product of two commuting Banach space operators with Dunford's property (C) have the single-valued extension property.     

Boolean Algebras,Generalized Abelian Rings,and Grothendieck Groups

ABSTRACT

A ring R is called generalized Abelian if for each idempotent e in R, eR and (1 ? e)R have no isomorphic nonzero summands. The class of generalized Abelian rings properly contains the class of Abelian rings. We denote by GAERS ? 1 the class of generalized Abelian exchange rings with stable range 1. In this article we prove, by introducing Boolean algebras, that for any R ∈ GAERS ? 1, the Grothendieck group K ₀(R) is always an Archimedean lattice-ordered group, and hence is torsion free and unperforated, which generalizes the corresponding results of Abelian exchange rings. Our main technical tool is the use of the ordered structure of K ₀(R)⁺, which provides a new method in the study of Grothendieck groups.     

交换群层的正合序列与同态定理

在交换群层（简称层）中给出了层的单（满）同态核与上核的泛性定理及正合交换图的一系列结论,进而证明了交换群层的同态（同构）定理。     

Free product actions with relative property (T) and trivial outer automorphism groups

We show that every non-amenable free product of groups admits free ergodic probability measure preserving actions which have relative property (T) in the sense of S. Popa (2006) [Pop06, Def. 4.1]. There are continuum many such actions up to orbit equivalence and von Neumann equivalence, and they may be chosen to be conjugate to any prescribed action when restricted to the free factors. We exhibit also, for every non-amenable free product of groups, free ergodic probability measure preserving actions whose associated equivalence relation has trivial outer automorphisms group. This gives, in particular, the first examples of such actions for the free group on 2 generators.