On right-angled reflection groups in hyperbolic spaces

Abstract

We show that the right-angled hyperbolic polyhedra of finite volume in the hyperbolic space ${\mathbb{H}}^n$ may only exist if $n\le 14.$ We also provide a family of such polyhedra of dimensions $n=3,4,...,8$. We prove that for $n=3,4$ the members of this family have the minimal total number of hyperfaces and cusps among all hyperbolic right-angled polyhedra of the corresponding dimension. This fact is used in the proof of the main result.