Reflection subgroups of reflection groups

Let G be a discrete group generated by reflections in hyperbolic or Euclidean space, and let H G be a finite index reflection subgroup. Suppose that the fundamental chamber of G is a finite volume polytope with k facets. We prove that the fundamental chamber of H has at least k facets.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 90–92, 2004Original Russian Text Copyright © by A. A. Felikson and P. V. Tumarkin