Reflection subgroups of Euclidean reflection groups

Invariants of Euclidean Reflection Groups

Reflection subgroups of Coxeter groups

We use geometry of Davis complex of a Coxeter group to investigate finite index reflection subgroups of Coxeter groups. The main result is the following: if G is an infinite indecomposable Coxeter group and H ⊂ G is a finite index reflection subgroup then the rank of H is not less than the rank of G. This generalizes results of [7]. We also describe some properties of the nerves of the group an...

On finite index reflection subgroups of discrete reflection groups

Let G be a discrete group generated by reflections in hyperbolic or Euclidean space, and H ⊂ G be a finite index subgroup generated by reflections. Suppose that the fundamental chamber of G is a finite volume polytope with k facets. In this paper, we prove that the fundamental chamber of H has at least k facets. 1. Let X be hyperbolic space H, Euclidean space E or spherical space S . A polytope...

Reflection subgroups of odd-angled Coxeter groups

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.

Euclidean simplices generating discrete reflection groups

Let P be a convex polytope in the spherical space S, in the Euclidean space E, or in the hyperbolic space H. Consider the group GP generated by reflections in the facets of P . We call GP a reflection group generated by P . The problem we consider in this paper is to list polytopes generating discrete reflection groups. The answer is known only for some combinatorial types of polytopes. Already...