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 in X is called a Coxeter polytope if all its dihedral angles are integer parts of π. A convex polytope in X 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. The tiles of the decomposition are called fundamental polytopes. In this paper we show that if the polytope admitting a Coxeter decomposition has exactly k facets then the fundamental polytope has at most k facets (Theorem 1). In particular, consider a finite covolume discrete group G generated by reflections in hyperbolic or Euclidean space and a finite index reflection subgroup H ⊂ G. If the fundamental chamber of G has exactly k facets, then the fundamental chamber of H has at least k facets. The authors are grateful to E. B. Vinberg for useful discussions and remarks. 2. A polytope P ⊂ X is called acute-angled if its dihedral angles are less or equal to π 2 . The minimal plane containing a face of a polytope we call an extension of this face. The proof of Theorem 1 is based on the following fact proved by E. M. Andreev (see [1]): Let P ⊂ X be an acute-angled polytope. If two faces of P have no common points then the extensions of these faces have no common points. A convex polytope with fixed Coxeter decomposition will be denoted by P , the fundamental polytope of the decomposition will be denoted by F . A hyperplane H is called a mirror of the decomposition if it contains a facet of a fundamental polytope and contains no facet of P . Let Θ be a Coxeter decomposition of P . A dihedral angle of P formed up by facets α and β is called fundamental in the decomposition Θ if no mirror of Θ contains α ∩ β.