Commuting involution graphs for \tilde{A}_{n}

In this article we consider the commuting graphs of involution conjugacy classes in the affine Weyl group We show that where the graph is connected the diameter is at most 6. Received: 24 February 2005     

The No Gap Conjecture for tame hereditary algebras

The “No Gap Conjecture” of Brüstle–Dupont–Pérotin states that the set of lengths of maximal green sequences for hereditary algebras over an algebraically closed field has no gaps. This follows from a stronger conjecture that any two maximal green sequences can be “polygonally deformed” into each other. We prove this stronger conjecture for all tame hereditary algebras over any field, equivalently, for any acyclic tame skew-symmetrizable exchange matrix.     

A note on geometric constructions of bi-invariant orderings

We construct bi-invariant total orderings of residually torsion-free nilpotent groups by using Chen's iterated integrals. This construction can be seen as a generalization of the Magnus ordering of the free groups, and equivalent to the classical construction which uses an iteration of central extensions. Our geometric construction provides a connection between bi-orderings and the rational homotopy theory.     

Algebraic invariants for right-angled Artin groups

A finite simplicial graph Γ determines a right-angled Artin group G_Γ, with generators corresponding to the vertices of Γ, and with a relation υw=wυ for each pair of adjacent vertices. We compute the lower central series quotients, the Chen quotients, and the (first) resonance variety of G_Γ, directly from the graph Γ. Partially supported by NSF grant DMS-0311142.     

A new approach to the construction of subsystems of complex root systems

Dynkin has shown how subsystems of real root systems may be constructed. As the concept of subsystems of complex root systems is not as well developed as in the real case, in this paper we give an algorithm to classify the proper subsystems of complex proper root systems. Furthermore, as an application of this algorithm, we determine the proper subsystems of imprimitive complex proper root systems. These proper subsystems are useful in giving combinatorial constructions of irreducible representations of properly generated finite complex reflection groups.     

Homological algebra for affine Hecke algebras

In this paper we study homological properties of modules over an affine Hecke algebra H. In particular we prove a comparison result for higher extensions of tempered modules when passing to the Schwartz algebra S, a certain topological completion of the affine Hecke algebra. The proof is self-contained and based on a direct construction of a bounded contraction of certain standard resolutions of H-modules.This construction applies for all positive parameters of the affine Hecke algebra. This is an important feature, since it is an ingredient to analyse how the irreducible discrete series representations of H arise in generic families over the parameter space of H. For irreducible non-simply laced affine Hecke algebras this will enable us to give a complete classification of the discrete series characters, for all positive parameters (we will report on this application in a separate article).     

Minimal length elements in some double cosets of Coxeter groups

We study the minimal length elements in some double cosets of Coxeter groups and use them to study Lusztig's G-stable pieces and the generalization of G-stable pieces introduced by Lu and Yakimov. We also use them to study the minimal length elements in a conjugacy class of a finite Coxeter group and prove a conjecture in [M. Geck, S. Kim, G. Pfeiffer, Minimal length elements in twisted conjugacy classes of finite Coxeter groups, J. Algebra 229 (2) (2000) 570-600].     

Quantum affine Cartan matrices,Poincaré series of binary polyhedral groups,and reflection representations

We first review some invariant theoretic results about the finite subgroups of SU(2) in a quick algebraic way by using the McKay correspondence and quantum affine Cartan matrices. By the way it turns out that some parameters (a, b, h; p, q, r) that one usually associates with such a group and hence with a simply-laced Coxeter–Dynkin diagram have a meaningful definition for the non-simply-laced diagrams, too, and as a byproduct we extend Saito’s formula for the determinant of the Cartan matrix to all cases. Returning to invariant theory we show that for each irreducible representation i of a binary tetrahedral, octahedral, or icosahedral group one can find a homomorphism into a finite complex reflection group whose defining reflection representation restricts to i.     

A (very short) introduction to buildings

This is an informal elementary introduction to buildings—what they are and where they come from.     

Classifying spaces with virtually cyclic stabilisers for certain infinite cyclic extensions

Let G?B?Z be an infinite cyclic extension of a group B where the action of Z on the set of conjugacy classes of non-trivial elements of B is free. This class of groups includes certain strictly descending HNN-extensions with abelian or free base groups, certain wreath products by Z and the soluble Baumslag-Solitar groups BS(1,m) with |m|≥2. We construct a model for , the classifying space of G for the family of virtually cyclic subgroups of G, and give bounds for the minimal dimension of . Finally we determine the geometric dimension when G is a soluble Baumslag-Solitar group.     

Reflection groups of geodesic spaces and Coxeter groups

H.S.M. Coxeter showed that a group Γ is a finite reflection group of an Euclidean space if and only if Γ is a finite Coxeter group. In this paper, we define reflections of geodesic spaces in general, and we prove that Γ is a cocompact discrete reflection group of some geodesic space if and only if Γ is a Coxeter group.     

CAT(0) groups and Coxeter groups whose boundaries are scrambled sets

In this paper, we study CAT(0) groups and Coxeter groups whose boundaries are scrambled sets. Suppose that a group G acts geometrically (i.e. properly and cocompactly by isometries) on a proper CAT(0) space X. (Such a group G is called a CAT(0) group.) Then the group G acts by homeomorphisms on the boundary ∂X of X and we can define a metric d_∂X on the boundary ∂X. The boundary ∂X is called a scrambled set if, for any α,β∈∂X with α≠β, (1) lim sup{d_∂X(gα,gβ)∣g∈G}>0 and (2) lim inf{d_∂X(gα,gβ)∣g∈G}=0. We investigate when boundaries of CAT(0) groups (and Coxeter groups) are scrambled sets.     

DIMENSION DEFECTS AND THE HOPF INVARIANT

Abstract

The concept of dimension defect of a mapping was introduced by H. Hopf in [5]. We generalize and answer questions about mappings S³ → S² which he raised at the end of that paper. Our main result is that a mapping S^2n-1 → Sⁿ with non-vanishing Hopf invariant does not have dimension defect.     

Embedding of two-colored right-angled Coxeter groups into products of two binary trees

We prove that every finitely generated 2-colored right-angled Coxeter group Γ can be quasi-isometrically embedded into the product of two binary trees. Moreover we show that the natural extension of this embedding to n-colored groups is not for every group quasi-isometric. Partially supported by Swiss National Science Foundation.     

Coxeter groups are virtually special

In this paper we prove that every finitely generated Coxeter group has a finite index subgroup that is the fundamental group of a special cube complex. Some consequences include: Every f.g. Coxeter group is virtually a subgroup of a right-angled Coxeter group. Every word-hyperbolic Coxeter group has separable quasiconvex subgroups.