Commentarii Mathematici Helvetici

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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, GSP-1, MOSCOW, RUSSIAN FEDERATION
(2) U.F.R. de Mathématiques, Université 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