# Commentarii Mathematici Helvetici

Full-Text PDF (149 KB) | Metadata | Table of Contents | CMH summary

**Volume 80, Issue 1, 2005, pp. 63–73**

**DOI: 10.4171/CMH/4**

On right-angled reflection groups in hyperbolic spaces

Igor M. Nikonov^{[1]}and Leonid Potyagailo

^{[2]}(1) Department of Mechanics and Mathematics, Moscow State University, 1 Leninskiye Gory, 119991, Moscow, Russian Federation

(2) U.F.R. de MathÃ©matiques, UniversitÃ© Lille I, 59655, Villeneuve-d'Ascq CEDEX, France

We show that the right-angled hyperbolic polyhedra of finite volume in the hyperbolic space $\Bbb H^n$ may only exist if $n\leq 14.$ 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.

*Keywords: *right-angled Coxeter polyhedra, reflection groups, hyperbolic orbifolds

Nikonov Igor, Potyagailo Leonid: On right-angled reflection groups in hyperbolic spaces. *Comment. Math. Helv.* 80 (2005), 63-73. doi: 10.4171/CMH/4