JournalscmhVol. 80 , No. 1DOI 10.4171/cmh/4

On right-angled reflection groups in hyperbolic spaces

  • Leonid Potyagailo

    Université Lille I, Villeneuve d'Ascq, France
  • Ernest Vinberg

    Moscow State University, Russia
On right-angled reflection groups in hyperbolic spaces cover

Abstract

We show that the right-angled hyperbolic polyhedra of finite volume in the hyperbolic space Hn\Bbb H^n may only exist if n14.n\leq 14. We also provide a family of such polyhedra of dimensions n=3,4,...,8n=3,4,...,8. We prove that for n=3,4n=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.