Extremal Subgroups in Chevalley Groups

Throughout this paper G(k) denotes a Chevalley group of rankn defined over the field k, where n3. Let be the root systemassociated with G(k) and let ={₁, ₂, ..., _n} be a set of fundamentalroots of , with ⁺ being the set of positive roots of with respectto . For and ⁺, let n() be the coefficient of in the expressionof as a sum of fundamental roots; so =n(). Also we recall thatht(), the height of , is given by ht()=n(). The highest rootin ⁺ will be denoted by . We additionally assume that the Dynkindiagram of G(k) is connected.     

On the Total Coloring of Graphs Embeddable in Surfaces

The paper shows that any graph G with the maximum degree (G) 8, which is embeddable in a surface of Euler characteristic() 0, is totally ((G)+2)-colorable. In general, it is shownthat any graph G which is embeddable in a surface and satisfiesthe maximum degree (G) (20/9) (3–())+1 is totally ((G)+2)-colorable.     

An Access Theorem for Continuous Functions

Let f be a continuous function on an open subset of R² suchthat for every x there exists a continuous map : [–1,1] with (0) = x and f increasing on [–1, 1]. Thenfor every there exists a continuous map : [0, 1) suchthat (0) = y, f is increasing on [0; 1), and for every compactsubset K of , max{t : (t) K} < 1. This result gives an answerto a question posed by M. Ortel. Furthermore, an example showsthat this result is not valid in higher dimensions.     

The Isolated Points of G{rho} and the w*-Strongly Exposed Points of P{rho}(G)0

Let G be a separable locally compact group and let be its dualspace with Fell's topology. It is well known that the set P(G)of continuous positive-definite functions on G can be identifiedwith the set of positive linear functionals on the group C^*-algebraC^*(G). We show that if is discrete in , then there exists anonzero positive-definite function associated with such that is a w^*-strongly exposed point of P(G)₀, where P(G)₀={f P(G):f(e)1. Conversely, if some nonzero positive-definite function associatedwith is a w^*-strongly exposed point of P(G)₀, then is isolatedin . Consequently, G is compact if and only if, for every ,there exists a nonzero positive-definite function associatedwith that is a w^*-strongly exposed point of P(G)₀. If, in addition,G is unimodular and , then is isolated in if and only if somenonzero positive-definite function associated with is a w^*-stronglyexposed point of P(G)₀, where is the left regular representationof G and is the reduced dual space of G. We prove that if B(G)has the Radon–Nikodym property, then the set of isolatedpoints of (so square-integrable if G is unimodular) is densein . It is also proved that if G is a separable SIN-group, thenG is amenable if and only if there exists a closed point in. In particular, for a countable discrete non-amenable groupG (for example the free group F₂ on two generators), there isno closed point in its reduced dual space .     

Livsic Regularity Theorems for Twisted Cocycle Equations Over Hyperbolic Systems

Let be a hyperbolic map. Cocycle equations of the form f =u·g·u^–1 are considered, with f, g, u takingvalues in a compact connected Lie group G, being an automorphismof G and f, g being Hölder continuous. When the eigenvaluesof the derivative of have modulus 1, it is proved that anymeasurable solution u has a Hölder continuous version.This condition on is optimal. When f, g are C^k then u may betaken to be C^k–1+ for any (0, 1).     

Movement and Separation of Subsets of Points Under Group Actions

Let G be a permutation group on a set , and let m and k be integerswhere 0<m<k. For a subset of , if the cardinalities ofthe sets ^g\, for gG, are finite and bounded, then is said tohave bounded movement, and the movement of is defined as move()=max_gG|^g\|. If there is a k-element subset such that move()m, it is shown that some G-orbit has length at most (k²–m)/(k–m).When combined with a result of P. M. Neumann, this result hasthe following consequence: if some infinite subset has boundedmovement at most m, then either is a G-invariant subset withat most m points added or removed, or nontrivially meets aG-orbit of length at most m²+m+1. Also, if move ()m for allk-element subsets and if G has no fixed points in , then either||k+m (and in this case all permutation groups on have thisproperty), or ||5m–2. These results generalise earlierresults about the separation of finite sets under group actionsby B. J. Birch, R. G. Burns, S. O. Macdonald and P. M. Neumann,and groups in which all subsets have bounded movement (by theauthor).     

On the Edge Reconstruction of Graphs Embedded in Surfaces II

In this paper, we prove the following theorems. (i) Let G bea graph of minimum degree 5. If G is embeddable in a surface and satisfies (–5)|V(G)|+6()0, then G is edge reconstructible.(ii) Any graph of minimum degree 4 that triangulates a surfaceis edge reconstructible. (iii) Any graph which triangulatesa surface of characteristic 0 is edge reconstructible.     

Higher Generation Subgroup Sets and the Virtual Cohomological Dimension of Graph Products of Finite Groups

We introduce panels of stabilizer schemes (K, G_*) associatedwith finite intersection-closed subgroup sets of a given groupG, generalizing in some sense Davis' notion of a panel structureon a triangulated manifold for Coxeter groups. Given (K, G_*),we construct a G-complex X with K as a strong fundamental domainand simplex stabilizers conjugate to subgroups in . It turnsout that higher generation properties of in the sense of Abels-Holzare reflected in connectivity properties of X. Given a finite simplicial graph and a non-trivial group G()for every vertex of , the graph product G() is the quotientof the free product of all vertex groups modulo the normal closureof all commutators [G(), G(w)] for which the vertices , w areadjacent. Our main result allows the computation of the virtualcohomological dimension of a graph product with finite vertexgroups in terms of connectivity properties of the underlyinggraph .     

The Bestvina-Brady Construction Revisited: Geometric Computation of {sum}-Invariants for Right-Angled Artin Groups

The starting point of our investigation is the remarkable paper[2] in which Bestvina and Brady gave an example of an infinitelyrelated group of type FP₂. The result about right-angled Artingroups behind their example is best interpreted by means ofthe Bieri–Strebel–Neumann–Renz -invariants. For a group G the invariants ⁿ(G) and ⁿ(G, Z) are sets of non-trivialhomomorphisms :GR. They contain full information about finitenessproperties of subgroups of G with abelian factor groups. Themain result of [2] determines for the canonical homomorphism, taking each generator of the right-angled Artin group G to1, the maximal n with ⁿ(G), respectively ⁿ(G, Z). In [6] Meier, Meinert and VanWyk completed the picture by computingthe full -invariants of right-angled Artin groups using as wellthe result of Bestvina and Brady as algebraic techniques from-theory. Here we offer a new account of their result which istotally geometric. In fact, we return to the Bestvina–Bradyconstruction and simplify their argument considerably by bringinga more general notion of links into play. At the end of thefirst section we re-prove their main result. By re-computingthe full -invariants, we show in the second section that thesimplification even adds some power to the method. The criterionwe give provides new insight on the geometric nature of the‘n-domination’ condition employed in [6].     

Faithful Representations of Free Products

In 1940 Nisnevi published the following theorem [3]. Let (G) be a family of groups indexed by some set and (F) a family of fields of the same characteristic p0. Iffor each the group G has a faithful representation of degreen over F then the free product* G has a faithful representationof degree n+1 over some field of characteristic p. In [6] Wehrfritzextended this idea. If (G) GL(n, F) is a family of subgroupsfor which there exists ZGL(n, F) such that for all the intersectionGF.1_n=Z, then the free product of the groups *_ZG with Z amalgamatedvia the identity map is isomorphic to a linear group of degreen over some purely transcendental extension of F. Initially, the purpose of this paper was to generalize theseresults from the linear to the skew-linear case, that is, togroups isomorphic to subgroups of GL(n, D) where the D are divisionrings. In fact, many of the results can be generalized to ringswhich, although not necessarily commutative, contain no zero-divisors.We have the following.     

Constructing {kappa}-like Models of Arithmetic

A model (M, <, ...) is -like if M has cardinality but, forall M, the cardinality of {x M : x < a} is strictly lessthan . In this paper we shall give constructions of -like modelsof arithmetic satisfying an arbitrarily large finite part ofPA but not PA itself, for various singular cardinals . The mainresults are: (1) for each countable nonstandard M ₂–Th(PA)with arbitrarily large initial segments satisfying PA and eachuncountable of cofinality there is a cofinal extension K ofM which is -like; also hierarchical variants of this resultfor _n–Th(PA); and (2) for every n 1, every singular and every M B_n+exp+¬ I_n there is a -like model K elementarilyequivalent to M.     

Recurrence, Dimension and Entropy

Let (_A, T) be a topologically mixing subshift of finite typeon an alphabet consisting of m symbols and let :_A R^d be a continuousfunction. Denote by (x) the ergodic limit when the limit exists. Possible ergodic limits arejust mean values dµ for all T-invariant measures. Forany possible ergodic limit , the following variational formulais proved: where h_µ denotes the entropy of µ and h_top denotestopological entropy. It is also proved that unless all pointshave the same ergodic limit, then the set of points whose ergodiclimit does not exist has the same topological entropy as thewhole space _A     

Real Interpolation and Two Variants of Gehring's Lemma

Let be a fixed open cube in Rⁿ. For r[1, ) and [0, ) we define where Q is a cube in Rⁿ (with sides parallel to the coordinateaxes) and _Q stands for the characteristic function of the cubeQ. A well-known result of Gehring [5] states that if (1.1) for some p(1, ) and c(0, ), then there exist q(p, ) and C=C(p,q, n, c)(0, ) such that for all cubes Q, where |Q| denotes the n-dimensional Lebesguemeasure of Q. In particular, a function fL¹() satisfying (1.1)belongs to L^q(). In [9] it was shown that Gehring's result is a particular caseof a more general principle from the real method of interpolation.Roughly speaking, this principle states that if a certain reversedinequality between K-functionals holds at one point of an interpolationscale, then it holds at other nearby points of this scale. Usingan extension of Holmstedt's reiteration formulae of [4] andresults of [8] on weighted inequalities for monotone functions,we prove here two variants of this principle involving extrapolationspaces of an ordered pair of (quasi-) Banach spaces. As an applicationwe prove the following Gehring-type lemmas.     

Geometry of Critical Loci

Let :(Z,z)(U,0) be the germ of a finite (that is, proper with finite fibres)complex analytic morphism from a complex analytic normal surfaceonto an open neighbourhood U of the origin 0 in the complexplane C². Let u and v be coordinates of C² defined on U. Weshall call the triple (, u, v) the initial data. Let stand for the discriminant locus of the germ , that is,the image by of the critical locus of . Let ()_A be the branches of the discriminant locus at O whichare not the coordinate axes. For each A, we define a rational number d by where I(–, –) denotes the intersection number at0 of complex analytic curves in C². The set of rational numbersd, for A, is a finite subset D of the set of rational numbersQ. We shall call D the set of discriminantal ratios of the initialdata (, u, v). The interesting situation is when one of thetwo coordinates (u, v) is tangent to some branch of , otherwiseD = {1}. The definition of D depends not only on the choiceof the two coordinates, but also on their ordering. In this paper we prove that the set D is a topological invariantof the initial data (, u, v) (in a sense that we shall definebelow) and we give several ways to compute it. These resultsare first steps in the understanding of the geometry of thediscriminant locus. We shall also see the relation with thegeometry of the critical locus.     

Character Quotients for Coprime Acting Groups

Let the finite group A be acting on a finite group G with (|A|,|G|)=1. Let be the semidirect product of A and G. Let be acharacter of irreducible after restriction to G. In a previouspaper by Brian Hartley and the author, we proved that the restrictionof to S belongs to the set C(S) obtained by running over all that arise in this manner, by assuming, in addition, that Gis a product of extraspecial groups. This was proved in general,assuming only some condition on the Green functions of groupsof Lie type that is not as yet fully verified. In the presentpaper, we define the map Q(): SC by Q()(s)=|C_G(s)|/(s). We provethat Q()C(S) under the same hypotheses. In particular, the characterquotient Q() is an ordinary character.     

Some Remarks on the Cone of Completely Positive Maps between Von Neumann Algebras

The close relationship between the notions of positive formsand representations for a C*-algebra A is one of the most basicfacts in the subject. In particular the weak containment ofrepresentations is well understood in terms of positive forms:given a representation of A in a Hilbert space H and a positiveform on A, its associated representation is weakly containedin (that is, ker ker ) if and only if belongs to the weak*closure of the cone of all finite sums of coefficients of .Among the results on the subject, let us recall the followingones. Suppose that A is concretely represented in H. Then everypositive form on A is the weak* limit of forms of the typex ^k_{i=1 i}, x_i with the _i in H; moreover if A is a von Neumannsubalgebra of (H) and is normal, there exists a sequence (_i)_{i 1} in H such that (x) = _{i 1 i}, x_i for all x.     

Weighted Integral Inequalities for the Hardy Type Operator and the FractionalMaximal Operator

Let w(x), u(x) and (x) be weight functions. In this paper, underappropriate conditions on Young's functions ₁, ₂ we characterizethe inequality for the Hardy-typeoperator T defined in [1] and the inequality for the fractional maximal operator M_{, ;} definedin [8], as well as the corresponding weak-type inequalities.     

Immanant Inequalities, Induced Characters, and Rank Two Partitions

If = {₁, ₂, ..., _s}, where _{1 2} ... _s > 0, is a partitionof n then denotes the associated irreducible character of S_n,the symmetric group on {1, 2, ..., n}, and, if cCS_n, the groupalgebra generated by C and S_n, then d_c(·) denotes thegeneralized matrix function associated with c. If c₁, c₂ CS_nthen we write c₁ c₂ in case (A) (A) for each n x n positivesemi-definite Hermitian matrix A. If cCS_n and c(e) 0, wheree denotes the identity in S_n, then or denotes (c(e))^–1 c. The main result, an estimate for the norms of tensors of a certainanti-symmetry type, implies that if = {₁, ₂, ..., _s, 1^t} isa partition of n such that s > 1 and _s = 2, and ' denotes{₁, ₂, ..., _s-1, 1^t} then (, _{2}) where denotes characterinduction from S_n–2 x S₂ to S_n. This in turn implies thatif = {₁, ₂, ..., _s, 1^t} with s > 1, _s = 2, and ßdenotes {₁ + 2, ₂, ..., _s-1, 1^t} then _ß which,in conjunction with other known results, provides many new inequalitiesamong immanants. In particular it implies that the permanentfunction dominates all normalized immanants whose associatedpartitions are of rank 2, a result which has proved elusivefor some years. We also consider the non-relationship problem for immanants– that is the problem of identifying pairs, (,ß)such that _ß and _ß are both false.     

A Class of Functional Equations on a Locally Compact Group

Let G be a locally compact group not necessarily unimodular.Let µ be a regular and bounded measure on G. We study,in this paper, the following integral equation, E(µ) This equation generalizes the functional equation for sphericalfunctions on a Gel'fand pair. We seek solutions in the spaceof continuous and bounded functions on G. If is a continuousunitary representation of G such that (µ) is of rank one,then tr((µ)(x)) is a solution of E(µ). (Here, trmeans trace). We give some conditions under which all solutionsare of that form. We show that E(µ) has (bounded and)integrable solutions if and only if G admits integrable, irreducibleand continuous unitary representations. We solve completelythe problem when G is compact. This paper contains also a listof results dealing with general aspects of E(µ) and propertiesof its solutions. We treat examples and give some applications.