On ADEG-polyhedra in hyperbolic spaces

TL;DR

This paper classifies ADEG-polyhedra in hyperbolic spaces, showing dihedral angles are restricted in large dimensions and identifying all such polyhedra with specific angles, including a new 9-dimensional example.

Contribution

It provides a complete classification of ADEG-polyhedra with specific dihedral angles and introduces a new polyhedron in nine-dimensional hyperbolic space.

Findings

Dihedral angles in high-dimensional Coxeter polyhedra are limited to specific fractions of π.

All ADEG-polyhedra with prescribed angles are classified, including three exceptional cases.

A new 9-dimensional polyhedron with 14 facets is constructed and described.

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 $n\geq 32$, they are of the form $\frac{\pi}{m}$ with $m\leq 6$. Moreover, this property holds in all dimensions $n\geq 7$ 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.