Taut Foliations in Punctured Surface Bundles, II

Given a fibered 3-manifold M, we investigate exactly which boundaryslopes can be realized by perturbing fibrations along productdiscs. Since these perturbed fibrations cap off to give tautfoliations in the corresponding surgery manifolds, we obtainsurgery information. For example, recall that a knot k is saidto satisfy Property P if no finite surgery along k yields asimply-connected 3-manifold. We show that if a non-trivial fiberedknot k S³ fails to satisfy Property P, then necessarily k ishyperbolic with degeneracy slope . When k is hyperbolic and (respectively, ), we show that the only candidate for a counterexample to Property P is surgery coefficient (respectively, . 2000 Mathematical Subject Classification: primary57M25; secondary 57R30.     

Property (T) for C*-Algebras

A notion of Property (T) is defined for an arbitrary unitalC*-algebra A admitting a tracial state. This is extended toa notion of Property (T) for a pair (A, B) where B is a C*-subalgebraof A. Let be a discrete group and its reduced algebra. We show that has Property (T) if and only if the group has Property (T).More generally, given a subgroup of , the pair has Property (T) if and only if the pair of groups(, ) has Property (T). 2000 Mathematics Subject Classification46L05, 22D25.     

A Sequential Algorithm for an Inventory Selection Problem

Assume that we have m finished products in an inventory. Eachfinished product is characterized by two measurements P andQ. A customer specifies a purchase order by the requirementsof characteristics P and Q. A product is qualified to satisfya purchase order if and only if it possesses better measurementsof both P and Q than the customer requires. For a given batchof n purchase orders, the inventory selection problem is tochoose n finished products from the inventory to satisfy allpurchase orders with a minimum cost. This problem can be formulatedas a large-scale transportation problem. When the cost functionof selecting a product to satisfy an order exhibits certainstructure, we develop a fast sequential algorithm to solve thisproblem. Possible extensions and related problems are also discussedin this paper.     

On the Algebra of Operations for Hopf Cohomology

In his thesis (Mem. Amer. Math. Soc. 42 (1962)) A. Liuleviciusdefined Steenrod squaring operations Sq^k on the cohomology ringof any cocommutative Hopf algebra over Z/2. Later, J. P. Mayshowed that these operations satisfy Adem relations, interpretedso that Sq⁰ is not the unit but an independent operation. Thus,these Adem relations are homogeneous of length two in the generators.This paper is concerned with the bigraded algebra B that isgenerated by elements Sq^k and subject to Adem relations; itshows that the Cartan formula gives a well-defined coproducton B. Also, it is shown that B with both multiplication andcomultiplication should be considered neither a Hopf algebranor a bialgebra, but another kind of structure, for which thename ‘algebra with coproducts’ is proposed in thepaper. 2000 Mathematics Subject Classification 55S10 (primary),55T15 (secondary).     

Group Laws and Free Subgroups in Topological Groups

A proof is given that a permutation group in which differentfinite sets have different stabilizers cannot satisfy any grouplaw. For locally compact topological groups with this property,almost all finite subsets of the group are shown to generatefree subgroups. Consequences of these theorems are derived for:Thompson's group F, weakly branch groups, automorphism groupsof regular trees, and profinite groups with alternating compositionfactors of unbounded degree. 2000 Mathematics Subject Classification20B07, 20E10, 20E18, 20P05.     

Duality theorems for convex problems of Lagrange with time-delay

Convex problems of Lagrange with time-delay in the state areexamined. The state x(t) is constrained in a convex set X(t)for each t. Optimal trajectories satisfy certain subdifferentialconditions which involve ‘multipliers’ that solvesome dual, convex problem.     

Locally Nilpotent p-Groups whose Proper Subgroups are Hypercentral or Nilpotent-by-Chernikov

Let G be a group and P be a property of groups. If every propersubgroup of G satisfies P but G itself does not satisfy it,then G is called a minimal non-P group. In this work we studylocally nilpotent minimal non-P groups, where P stands for ‘hypercentral’or ‘nilpotent-by-Chernikov’. In the first case weshow that if G is a minimal non-hypercentral Fitting group inwhich every proper subgroup is solvable, then G is solvable(see Theorem 1.1 below). This result generalizes [3, Theorem1]. In the second case we show that if every proper subgroupof G is nilpotent-by-Chernikov, then G is nilpotent-by-Chernikov(see Theorem 1.3 below). This settles a question which was consideredin [1–3, 10]. Recently in [9], the non-periodic case ofthe above question has been settled but the same work containsan assertion without proof about the periodic case. The main results of this paper are given below (see also [13]).     

Perfect and Acyclic Subgroups of Finitely Presentable Groups

Acyclic groups of low dimension are considered. To indicatethe results simply, let G' be the nontrivial perfect commutatorsubgroup of a finitely presentable group G. Then def(G)1. Whendef(G)=1, G' is acyclic provided that it has no integral homologyin dimensions above 2 (a sufficient condition for this is thatG' be finitely generated); moreover, G/G' is then Z or Z². Naturalexamples are the groups of knots and links with Alexander polynomial1. A further construction is given, based on knots in S²x S¹.In these geometric examples, G' cannot be finitely generated;in general, it cannot be finitely presentable. When G is a 3-manifoldgroup it fails to be acyclic; on the other hand, if G' is finitelygenerated it has finite index in the group of a Q-homology 3-sphere.     

On the Automorphism Groups of Cayley Graphs of Finite Simple Groups

Let G be a finite nonabelian simple group and let be a connectedundirected Cayley graph for G. The possible structures for thefull automorphism group Aut are specified. Then, for certainfinite simple groups G, a sufficient condition is given underwhich G is a normal subgroup of Aut. Finally, as an applicationof these results, several new half-transitive graphs are constructed.Some of these involve the sporadic simple groups G = J₁, J₄,Ly and BM, while others fall into two infinite families andinvolve the Ree simple groups and alternating groups. The twoinfinite families contain examples of half-transitive graphsof arbitrarily large valency.     

On the Geometry of the Automorphism Group of a Free Group

The groups Aut(F₃) and Out(F₃) satisfy strictly exponentialisoperimetric inequalities; in particular, they are not automatic.For n 3, Aut (F_n) and Out (F_n) do not admit bounded bicombingsof sub-exponential length, hence they cannot act properly andcocompactly by isometries on any simply-connected space of non-positivecurvature, and they are not biautomatic.     

On Semidirect Products and the Arithmetic Lifting Property

Let G be a finite group and let K be a hilbertian field. Manyfinite groups have been shown to satisfy the arithmetic liftingproperty over K, that is, every G-Galois extension of K arisesas a specialization of a geometric branched covering of theprojective line defined over K. The paper explores the situationwhen a semidirect product of two groups has this property. Inparticular, it shows that if H is a group that satisfies thearithmetic lifting property over K and A is a finite cyclicgroup then G = A H also satisfies the arithmetic lifting propertyassuming the orders of H and A are relatively prime and thecharacteristic of K does not divide the order of A. In thiscase, an arithmetic lifting for any AH-Galois extension of Kis explicitly constructed and the existence of the arithmeticlifting for any G-Galois extension is deduced. It is also shownthat if A is any abelian group, and H is the group with thearithmetic lifting property then AH satisfies the property aswell, with some assumptions on the ground field K. In the constructionproperties of Hilbert sets in hilbertian fields and spectralsequences in étale cohomology are used.     

A Characterisation of p-Soluble Groups

If p is a prime, then a finite group is p-soluble if each ofits composition factors is either a p-group or has order coprimeto p. For example, soluble groups are p-soluble. However, thereare many insoluble groups that are p-soluble. We shall provethe following result. 1991 Mathematics Subject Classification20D10.     

A new approach to energy-based sparse finite-element spaces

^{We show that the logarithmic factor in the standard error estimatefor sparse finite element (FE) spaces in arbitrary dimensiond is removable in the energy (H1) norm. Via a penalized sparsegrid condition, we then propose and analyse a new version ofthe energy-based sparse FE spaces introduced first in Bungartz(1992, Dünne Gitter und deren Anwendung bei der adaptivenLösung der dreidimensionalen Poisson-Gleichung. Dissertation.Munich, Germany: TU München) and known to satisfy an optimalapproximation property in the energy norm.}     

The spine of a Fourier-Stieltjes algebra

We define the spine A *(G) of the Fourier–Stieltjes algebraB (G) of a locally compact group G. This algebra encodes informationabout much of the fine structure of B (G), particularly informationabout certain homomorphisms and idempotents. We show that A *(G) is graded over a certain semi-lattice, thatof non-quotient locally precompact topologies on G. We computethe spine's spectrum G*, which admits a semi-group structure.We discuss homomorphisms from A *(G) to B (H) where H is anotherlocally compact group; and we show that A *(H) contains theimage of every completely bounded homomorphism from the Fourieralgebra A (H) of any amenable group G. We also show that A *(G)contains all of the idempotents in B (G). Finally, we computeexamples for vector groups, abelian lattices, minimally almostperiodic groups and the (ax + b)-group; and we explore the complexityof A *(G) for the discrete rational numbers and free groups.     

Semisimple Modules for Finite Groups of Lie Type

Let k be an algebraically closed field of characteristic p >0, and let G be a connected, reductive algebraic group overk. In [8] and [11], conditions on the dimension of rationalG modules were seen to imply semisimplicity of these modules.In [8], certain of these conditions were extended to cover thefinite groups of Lie type. In this paper, we extend some ofthe results of [11] to cover these finite Lie type groups. Themain such extension is the following result.     

Acyclic Edge Coloring of Triangle‐Free Planar Graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a′(G) ? Δ + 2, where Δ = Δ(G) denotes the maximum degree of the graph. If every induced subgraph H of G satisfies the condition |E(H)| ? 2|V(H)|?1, we say that the graph G satisfies Property A. In this article, we prove that if G satisfies Property A, then a′(G) ? Δ + 3. Triangle‐free planar graphs satisfy Property A. We infer that a′(G) ? Δ + 3, if G is a triangle‐free planar graph. Another class of graph which satisfies Property A is 2‐fold graphs (union of two forests). © 2011 Wiley Periodicals, Inc. J Graph Theory     

A strange example on Property Np

 We exhibit an example of a line bundle M on a smooth complex projective variety Y s.t. M satisfies Property N ₁₀ , the 10-module of a minimal resolution of the ideal of the embedding of Y by M is nonzero and M ² does not satisfy Property N ₁₀ . Thus this is a completely convincing example showing that surprisingly it is not true that if a line bundle M satisfies Property N _p then any power of M satisfies Property N _p . We recall that in [Ru] we proved the following statement: if M is a line bundle on a smooth complex projective variety and M satisfies Property N _p then M ^s satisfies Property N _p if s≥p. Received: 5 March 2001     

Primitive Representations by Spinor Genera of Ternary Quadratic Forms

Let a be primitively represented by the genus of a ternary quadraticlattice L defined over the ring of integers of an algebraicnumber field F. Criteria to determine whether a is primitivelyrepresented by every spinor genus in the genus of L involvecertain subgroups *(L_p, a) of the multiplicative groups of thelocalizations F_p of F with respect to the various nonarchimedeanprime spots p on F. In this paper these groups *(L_p, a) aredetermined explicitly for nondyadic and 2-adic prime spots.Examples are given which show how this information can, in someinstances, be used in combination with known results, to determineall integers primitively represented by a particular positivedefinite ternary quadratic form.     

Finite-element wavelets on manifolds

We construct locally supported, continuous wavelets on manifolds that are given as the closure of a disjoint union of generalsmooth parametric images of an n-simplex. The wavelets are provento generate Riesz bases for Sobolev spaces H^s () when s (–1,3/2), if not limited by the global smoothness of . These resultsgeneralize the findings from Dahmen & Stevenson (1999) SIAMJ. Numer. Anal., 37, 319–352, where it was assumed thateach parametrization has a constant Jacobian determinant. Thewavelets can be arranged to satisfy the cancellation propertyof, in principle, any order, except for wavelets with supportsthat extend to different patches, which generally satisfy thecancellation property of only order 1.     

Seminormal Representations of Weyl Groups and Iwahori-Hecke Algebras

The purpose of this paper is to describe a general procedurefor computing analogues of Young's seminormal representationsof the symmetric groups. The method is to generalize the Jucys-Murphyelements in the group algebras of the symmetric groups to arbitraryWeyl groups and Iwahori-Hecke algebras. The combinatorics ofthese elements allows one to compute irreducible representationsexplicitly and often very easily. In this paper we do thesecomputations for Weyl groups and Iwahori-Hecke algebras of typesA_n, B_n, D_n, G₂. Although these computations are in reach fortypes F₄, E₆ and E₇, we shall postpone this to another work.1991 Mathematics Subject Classification: primary 20F55, 20C15;secondary 20C30, 20G05.