We give a criterion for a finitely generated odd-angled Coxeter group to have a proper finite index subgroup generated by reflections. The answer is given in terms of the least prime divisors of the exponents of the Coxeter relations.

In this paper, we classify Coxeter decompositions of hyperbolic tetrahedra, i.e. simplices in the hyperbolic space IH. The paper together with [2] completes the classification of Coxeter decompositions of hyperbolic simplices.

In any Coxeter group, the conjugates of elements in the standard minimal generating set are called reflections and the minimal number of reflections needed to factor a particular element is called its reflection length. In this article we prove that the reflection length function on an affine Coxeter group has a uniform upper bound. More precisely we prove that the reflection length function on...

Given a Coxeter system (W, S), there is an associated CW-complex, denoted Σ(W, S) (or simply Σ), on which W acts properly and cocompactly. This is the Davis complex. L, the nerve of (W, S), is a finite simplicial complex. We prove that when (W, S) is an even Coxeter system and L is a flag triangulation of S, then the reduced `-homology of Σ vanishes in all but the middle dimension. In so doing,...

We give closed combinatorial product formulas for Kazhdan–Lusztig poynomials and their parabolic analogue of type q in the case of boolean elements, introduced in [M. Marietti, Boolean elements in Kazhdan–Lusztig theory, J. Algebra 295 (2006)], in Coxeter groups whose Coxeter graph is a tree. Such formulas involve Catalan numbers and use a combinatorial interpretation of the Coxeter graph of th...

Boyd (1974) proposed a class of infinite ball packings that are generated by inversions. Later, Maxwell (1983) interpreted Boyd’s construction in terms of root systems in Lorentz spaces. In particular, he showed that the space-like weight vectors correspond to a ball packing if and only if the associated Coxeter graph is of “level 2”. In Maxwell’s work, the simple roots form a basis of the repr...

In this work we introduce a new combinatorial notion of boundary <C of an ω-dimensional cubing C. <C is defined to be the set of almost-equality classes of ultrafilters on the standard system of halfspaces of C, endowed with an order relation reflecting the interaction between the Tychonoff closures of the classes. When C arises as the dual of a cubulation – or discrete system of halfspaces – H...

We generalize earlier work of Fuertes and González-Diez as well as earlier work of Bauer, Catanese and Grunewald by classifying which of the irreducible Coxeter groups are (strongly real) Beauville groups. We also make partial progress on the much more difficult question of which Coxeter groups are Beauville groups in general as well as discussing the related question of which Coxeter groups ca...

A coxeter matroid is a generalization of matroid, ordinary matroid being the case corresponding to the family of Coxeter groups An , which are isomorphic to the symmetric groups. A basic result in the subject is a geometric characterization of Coxeter matroid in terms of the matroid polytope, a result first stated by Gelfand and Serganova. This paper concerns properties of the matroid polytope....

The following results are proved: (1) The center of any finite index subgroup of an irreducible, infinite, non-affine Coxeter group is trivial; (2) Any finite index subgroup of an irreducible, infinite, non-affine Coxeter group cannot be expressed as a product of two nontrivial subgroups. These two theorems imply a unique decomposition theorem for a class of Coxeter groups. . MSC 2000 Subject C...