Boundaries of reduced free group C*-algebras

We prove that the crossed product C*-algebra C*_r(, ) of a freegroup with its boundary sits naturally between the reducedgroup C*-algebra C*_r and its injective envelope I(C*_r). In otherwords, we have natural inclusion C*_r C*_r(, ) I(C*_r) of C*-algebras.     

Trivial Cycles in Graphs Embedded in 3-Manifolds

A cycle C of a graph embedded in a 3-manifold M is said tobe trivial in if it bounds a disk with interior disjoint from. Let e be an edge of with ends on C. We shall study the relationbetween triviality of cycles in and that of – e and/e. Let C₁ be one of the two cycles in C e containing e. Themain theorem says that if C is trivial in – e and C₁/eis trivial in /e, then either C or C₁ is trivial in . Some applicationsto cycle trivial graphs will be given in Section 2.     

Non-Normal Pairs of Non-Euclidean Crystallographic Groups

Let be a non-Euclidean crystallographic group. is said tobe non-maximal if there exists a non-Euclidean crystallographicgroup ' such that ' and the dimension of the Teichmüllerspace of equals the dimension of the Teichmüller spaceof '. The full list of such pairs of groups is computed in thecase when is non-normal in '. The corresponding problem forFuchsian groups was solved by Singerman. 2000 Mathematics SubjectClassification 20H10 (primary), 30F10 (secondary).     

A Schwarz Lemma for the Symmetrized Bidisc

Let be an analytic function from D to the symmetrized bidisc We show that if (0) = (0,0) and () = (s, p) in the interiorof , then Moreover, the inequality is sharp: we give an explicit formulafor a suitable in the event that the inequality holds withequality. We show further that the inverse hyperbolic tangentof the left-hand side of the inequality is equal to both theCaratheodory distance and the Kobayashi distance from (0,0)to (s, p) in int      

Metric Entropy of Convex Hulls in Hilbert Spaces

We show in this note the following statement which is an improvementover a result of R. M. Dudley and which is also of independentinterest. Let X be a set of a Hilbert space with the propertythat there are constants , >0, and for each n N, the setX can be covered by at most n balls of radius n^–. Then,for each n N, the convex hull of X can be covered by 2ⁿ ballsof radius . The estimate is best possible for all n N, apart from the value c=c(, , X).In other words, let N(, X), >0, be the minimal number ofballs of radius covering the set X. Then the above result isequivalent to saying that if N(, X)=O(^–1/) as 0, thenfor the convex hull conv (X) of X, N(, conv (X)) =O(exp(^–2/(12))). Moreover, we give an interplay between several coveringparameters based on coverings by balls (entropy numbers) andcoverings by cylindrical sets (Kolmogorov numbers). 1991 MathematicsSubject Classification 41A46.     

An infinite Family of 4-Arc-Transitive Cubic Graphs Each with Girth 12

If p is any prime, and is that automorphism of the group SL(3,p) which takes each matrix to the transpose of its inverse,then there exists a connected trivalent graph (p) on vertices with the split extensionSL(3, p) as a group of automorphisms acting regularly on its4-arcs. In fact if p 3 then this group is the full automorphismgroup of (p), while the graph (3) is 5-arc-transitive with fullautomorphism group SL(3,3)0 x C₂. The girth of (p) is 12, exceptin th case p = 2 (where the girth is 6). Furthermore, in allcases (p) is bipartite, with SL(3, p) fixing each part. Alsowhen p 1 mod 3 the graph (p) is a triple cover of another trivalentgraph, which has automorphism group PSL(3, p)0 acting regularlyon its 4-arcs. These claims are proved using elementary theoryof symmetric graphs, together with a suitable choice of threematrices which generate SL(3, Z). They also provide a proofthat the group 4⁺(a¹²) described by Biggs in Computational grouptheor(ed. M. Atkinson) is infinite.     

Spaces Universal under Closed Embeddings for Finite-Dimensional Complete Metric Spaces

We shall prove that for every natural number n and every cardinalnumber there exists an n-dimensional complete metric spaceX_n, of weight such that every n-dimensional complete metricspace of weight is embeddable in X_n, as a closed subset.     

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 Variation on a Theorem of Galvin and Hajnal

Let be a singular cardinal of regular uncountable cofinality. Let {(): < } be a continuous increasing sequence withlimit , and let =₍₎₊₍₎, < be regular cardinals. Let I be a normal ideal on , and assume that the reduced product_</I admits a cofinal -scale of ordinal functions. Then ₊, where =||||_I is the I-norm of .     

{Omega}-Covers of Graphs

Given a group G, a G-set and a graph we present a constructionfor a family of graphs, the -covers of . A particular exampleof this construction gives a girth 17 cubic graph with 2530vertices. 2000 Mathematics Subject Classification 05C25, 05C35.     

Abelian Subgroups of Finitely Generated Kleinian Groups are Separable

By a Kleinian group we mean a discrete subgroup of PSL(2, C).We prove that abelian subgroups of finitely generated Kleiniangroups are separable. In other words, if M = H³/ is a hyperbolic3-orbifold, with finitely generated, then abelian subgroupsof are separable in . 1991 Mathematics Subject Classification20E26, 51M10, 57M05.     

Approximation of Ground State Eigenvalues and Eigenfunctions of Dirichlet Laplacians

Let be a bounded connected open set in R^N, N 2, and let –0be the Dirichlet Laplacian defined in L²(). Let > 0 be thesmallest eigenvalue of –, and let > 0 be its correspondingeigenfunction, normalized by ||||₂ = 1. For sufficiently small>0 we let R() be a connected open subset of satisfying Let – 0 be the Dirichlet Laplacian on R(), and let >0and >0 be its ground state eigenvalue and ground state eigenfunction,respectively, normalized by ||||₂=1. For functions f definedon , we let Sf denote the restriction of f to R(). For functionsg defined on R(), we let Tg be the extension of g to satisfying 1991 Mathematics SubjectClassification 47F05.     

Calculs De Facteurs Epsilon De Paires Pour GLn Sur Un Corps Local, I

Soient F un corps commutatif localement compact non archimédienet un caractère additif non trivial de F. Soient unereprésentation du groupe de Weil–Deligne de F,et sa contragrédiente. Nous calculons le facteur (, , ). De manière analogue, nous calculons le facteur (x, , ) pour toute représentationadmissible irréductible de GL_n(F). En conséquence,si F est de caractéristique nulle et si et se correspondentpar la correspondance de Langlands construite par M. Harris,ou celle construite par les auteurs, alors les facteurs (, , s) et (x, , s) sont égaux pour tout nombre complexe s. Let F be a non-Archimedean local field and a non-trivial additivecharacter of F. Let be a representation of the Weil–Delignegroup of F and its contragredient representation. We compute (, , ). Analogously, we compute (x, , ) for all irreducible admissible representations of GL_n(F).Consequently, if F has characteristic zero, and , correspondvia the Langlands correspondence established by M. Harris orthe correspondence constructed by the authors, then we have(, , s) = (x, , s) for all sC. 1991 Mathematics Subject Classification22E50.     

A Combinatorial Application of the Maximal Ergodic Theorem

Let X be a compact space,µ a Borel probability measureon X, T: X X a measure preserving continuous transformationand g: X R a continuous function. Then for some yX, This Lemma is used to give an alternative proof of a resultby Ruzsa [6], which implies the following extension of a resultof Bergelson [1]. If E N satisfies then there exists a set N such that n–1|[1,n]| (E) for all, n 1, and any finite subset{₁, ... _k} satisfies Ø. 7 Moria St., Ramat Hasharon, Israel     

Belyi Uniformization of Elliptic Curves

Belyi's Theorem implies that a Riemann surface X representsa curve defined over a number field if and only if it can beexpressed as U/, where U is simply-connected and is a subgroupof finite index in a triangle group. We consider the case whenX has genus 1, and ask for which curves and number fields canbe chosen to be a lattice. As an application, we give examplesof Galois actions on Grothendieck dessins. 1991 MathematicsSubject Classification 30F10, 11G05.     

On Transitive Permutation Groups with Primitive Subconstituents

Let G be a transitive permutation group on a set such that,for , the stabiliser G induces on each of its orbits in \{}a primitive permutation group (possibly of degree 1). Let Nbe the normal closure of G in G. Then (Theorem 1) either N factorisesas N=GG for some , , or all unfaithful G-orbits, if any exist,are infinite. This result generalises a theorem of I. M. Isaacswhich deals with the case where there is a finite upper boundon the lengths of the G-orbits. Several further results areproved about the structure of G as a permutation group, focussingin particular on the nature of certain G-invariant partitionsof . 1991 Mathematics Subject Classification 20B07, 20B05.     

On the Behaviour of the First Eigenfunction of the p-Laplacian Near its Critical Points

In this paper, the behaviour of the positive eigenfunction of in u_| = 0, p > 1, isstudied near its critical points. Under some convexity and symmetryassumptions on , is seen to have a unique critical point atx = 0; also, the behaviour of both and is determined nearby.Positive solutions u to some general problems –_pu = f(u)in , u_| = 0, are also considered, with some convexity restrictionson u. 2000 Mathematics Subject Classification 35B05 (primary),35J65, 35J70 (secondary).     

Are All Uniform Algebras Amnm?

A Banach algebra a is AMNM if whenever a linear functional on a and a positive number satisfy |(ab)–(a)(b)|||a||·||b||for all a, b a, there is a multiplicative linear functional on a such that ||–||=o(1) as 0. K. Jarosz [1] asked whetherevery Banach algebra, or every uniform algebra, is AMNM. B.E. Johnson [3] studied the AMNM property and constructed a commutativesemisimple Banach algebra that is not AMNM. In this note weconstruct uniform algebras that are not AMNM. 1991 MathematicsSubject Classification 46J10.     

The Gabriel-Roiter measure for radical-square zero algebras

Let be a radical-square zero algebra over an algebraicallyclosed field k with radical , and let be the associated hereditary algebra. Thereis an explicit functor F: mod mod , which induces a stableequivalence. In this paper, it will be proved that the functorF preserves the Gabriel–Roiter (GR) measures and the GRfactors. Thus the GR measure for can be studied by the useof F and known facts for hereditary algebras. In particular,the middle terms of the Auslander–Reiten sequences endingat the GR factors and the relationship between the preprojectivepartition for and the take-off -modules will be investigated.     

Cardinal Invariants and Eventually Different Functions

In this paper we study several kinds of maximal almost disjointfamilies. In the main result of this paper we show that forsuccessor cardinals , there is an unexpected connection betweeninvariants a_e(), b() and a certain cardinal invariant m_d(⁺)on ⁺. As a corollary we get for example the following result.For a successor cardinal , even assuming that ^< = and 2= ⁺, the following is not provable in Zermelo–Fraenkelset theory. There is a ⁺-cc poset which does not collapse andwhich forces a() = ⁺ < a_e() = ⁺⁺ = 2. We also apply the ideasfrom the proofs of these results to study a = a() and non(M).2000 Mathematics Subject Classification 03E17 (primary), 03E05(secondary).