On right-angled reflection groups in hyperbolic spaces

Abstract

We show that the right-angled hyperbolic polyhedra of finite volume in the hyperbolic space ℍn may only exist if n ≤ 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. © Swiss Mathematical Society.