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The normalizer $N_W(W_J)$ of a standard parabolic subgroup $W_J$ of a finite Coxeter group $W$ splits over the parabolic subgroup with complement $N_J$ consisting of certain minimal length coset representatives of $W_J$ in $W$. In this note we show that (with the exception of a small number of cases arising from a situation in Coxeter groups of type $D_n$) the centralizer $C_W(w)$ of an element $w \in W$ is in a similar way a semidirect product of the centralizer of $w$ in a suitable small parabolic subgroup $W_J$ with complement isomorphic to the normalizer complement $N_J$., final version, 18 pages, to appear in J. Group Theory

We provide a case-free characterization of hyperbolic Coxeter systems depending only on the Coxeter graph. Consequently, we uncover a case-free proof of the fact that every infinite, non-affine Coxeter system contains a standard parabolic subsystem that is a hyperbolic Coxeter system.

In this work we address the problem on the determination of the subgroup structure of crystallographic groups in hyperbolic space by deriving the low index subgroups of hyperbolic tetrahedral Coxeter groups and tetrahedral Kleinian groups. This paper continues the work giv en in [5, 6] on the subgroups of triangle groups.

In this paper we discuss some relations between (pre-)Coxeter algebras and its related topics, and obtain some equivalent conditions.

A convex polytope admits a Coxeter decomposition if it is tiled by finitely many Coxeter polytopes such that any two tiles having a common facet are symmetric with respect to this facet. In this paper, we classify all Coxeter decompositions of compact hyperbolic Coxeter n-polytopes with n + 2 facets. Furthermore, going out from Schläfli‘s reduction formula for simplices we construct in a purely combinatorial way a volume formula for arbitrary polytopes and compute the volumes of all compact Coxeter polytopes in ℍ4 which are products of simplices.

Suppose that W is an infinite Coxeter group of finite rank n, and suppose that W has a finite parabolic subgroup WJ of rank n � 1. Suppose also that the Coxeter diagram of W has no edges with infinite labels. Then any automorphism of W that preserves reflections lies in the subgroup of AutðW Þ generated by the inner automorphisms and the automorphisms induced by symmetries of the Coxeter graph. If, in addition, WJ is irreducible and every conjugacy class of reflections in W has nonempty intersection with WJ , then all automorphisms of W preserve reflections, and it follows that AutðW Þ is the semi-direct product of InnðW Þ by the group of graph automorphisms.

We determine the epimorphisms A --> W from the Artin group A of type Gamma onto the Coxeter group W of type Gamma, in case Gamma is an irreducible Coxeter graph of spherical type, and we prove that the kernel of the standard epimorphism is a characteristic subgroup of A. This generalizes an over 50 years old result of Artin

Given a finite group G and G-conjugacy class of involutions X, the local fusion graph F(G,X) has X as its vertex set, with x,y in X joined by an edge if, and only if, x is not equal to y and the product xy has odd order. In this note we investigate such graphs when G is a finite Coxeter group, addressing questions of connectedness and diameter. In particular, our results show that local fusion graphs may have an arbitrary number of connected components, each with arbitrarily large diameter.

Any normal reflection subgroup f W of a Coxeter system (W, S) is a factor in a semidirect product decomposition of W as described by Bonnafe and Dyer. Namely, S is the union of two subsets I and J such that no element of I is conjugate to an element of J , f W is the subgroup generated by WI conjugates of elements of J , and W is the semidirect product of WI by f W . This note describes the reduced expressions of elements of the form wxw−1 with w ∈ WI and x ∈ WJ in terms of reduced expressions of x and a suitable element of WI . Introduction and Preliminaries Following Bonnafe and Dyer, we consider a Coxeter system (W,S) in its internal semidirect product decomposition with respect to two subsets I, J ⊂ S such that I ∩J = ∅ and I ∪J = S and with the requirement that no element of I is conjugate to an element of J . The latter statement says that the vertex set of the Coxeter graph of (W,S) can be divided into two subsets with only edges with even label connecting the two. Let T represent the set of reflections of W . In [4] (see also [1]), it is shown that W is the semidirect product of WI and the normal subgroup W generated by WI -conjugates of elements of J . We observe that any normal reflection subgroup W of W arises from this type of decomposition of S. Then, we describe the reduced expressions of elements of the form wxw−1 where w ∈ WI and x ∈ WJ in terms of reduced expressions of w and x. From this, we get a slightly stronger result describing reduced expressions of reflections of the form wtw−1 with t ∈ WJ ∩ T and w ∈ WI . For general results about Coxeter systems consult [5]. 1. Statement of Main Result We begin with the same setup as [1]. We let (W,S) be a Coxeter system with ` : W → N the corresponding length function. Furthermore, assume that S = I ∪J with I ∩J = ∅ and that no element of I is conjugate to an element of J . We denote by WI and WJ the standard parabolic subgroups generated by I and J respectively. For any K, L ⊂ S, we let W := W/WK (respectively W := WL\W ) be the set of shortest right coset representatives of WK in W (respectively the set of shortest left coset representatives of WL in W ). Then we define W := W ∩ W to be the set of shortest representatives for the double cosets WL\W/WK in W . For brevity, we will use the following notation W K := WK ∩ W. Additionally, let T = ∪w∈W wSw−1 be the set of reflections of W . If P(T ) denotes the power set of Date: February 20, 2009. 1Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556; tedgar@nd.edu

For a finitely generated subgroup W ′ of a Coxeter system (W, S), there are finitely generated reflection subgroups W1, . . . , Wn of W , each containing W ′, such that any reflection subgroup of W containing W ′ contains one of the Wi as a standard parabolic subgroup. The canonical Coxeter generators of the Wi, and an expression for the parabolic closure of W ′ as a W -conjugate of a standard parabolic subgroup of W , may be effectively determined.

We provide conditions on the defining graph of a rightangled Coxeter group presentation that guarantees the boundary of any CAT (0) space on which the group acts geometrically will be locally

In Involution products in Coxeter groups [J. Group Theory 14 (2011), no. 2, 251–259], Theorem 2.4 states a well-known result on Coxeter groups which gives conditions under which the stabilizer of a nonzero vector is a proper parabolic subgroup. However the hypothesis of this result is incorrectly stated in our paper: it holds for finite Coxeter groups but is not true in general for infinite Coxeter groups. We are grateful to an anonymous referee of a subsequent paper for pointing this out. As a consequence, the proof of Theorem 1.1 in that paper, which uses Theorem 2.4, is incomplete. Here we complete the proof of Theorem 1.1 without recourse to Theorem 2.4.

We show that any finitely generated Coxeter group acts properly discontinuously on a locally finite, finite dimensional CAT(0) cube complex. For any word hyperbolic or right angled Coxeter group we prove that the cubing is cocompact. We show how the local structure of the cubing is related to the partial order studied by Brink and Howlett in their proof of automaticity for Coxeter groups.