On ADEG-polyhedra in hyperbolic spaces

Abstract

In this paper, we establish that the non-zero dihedral angles of hyperbolic Coxeter polyhedra of large dimensions are not arbitrarily small. Namely, for dimensions $n\ge 32$, they are of the form $\frac{\pi }{m}$ with $m\le 6$. Moreover, this property holds in all dimensions $n\ge 7$ for Coxeter polyhedra with mutually intersecting facets. Then, we develop a constructive procedure tailored to Coxeter polyhedra with prescribed dihedral angles, from which we derive the complete classification of ADEG-polyhedra, characterized by having no pair of disjoint facets and dihedral angles $\frac{\pi }{2},\frac{\pi }{3}$ and $\frac{\pi }{6}$, only. Besides some well-known simplices and pyramids, there are three exceptional polyhedra, one of which is a new polyhedron $P_{\star }\subset {\mathbb{H}}^9$ with $14$ facets.

50 pages