Constructing Hyperbolic Polyhedra Using Newton's Method

Abstract

We demonstrate how to construct three-dimensional compact hyperbolic polyhedra using Newton's method. Under the restriction that the dihedral angles must be nonobtuse, Andreev's theorem [Andreev 70a, Andreev 70b] provides as necessary and sufficient conditions five classes of linear inequalities for the dihedral angles of a compact hyperbolic polyhedron realizing a given combinatorial structure C. Andreev's theorem also shows that the resulting polyhedron is unique, up to hyperbolic isometry. Our construction uses Newton's method and a homotopy to explicitly follow the existence proof presented by Andreev, providing both a very clear illustration of a proof of Andreev's theorem and a convenient way to construct three-dimensional compact hyperbolic polyhedra having nonobtuse dihedral angles. As an application, we construct compact hyperbolic polyhedra having dihedral angles that are (proper) integer submultiples of π, so that the group Γ generated by reflections in the faces is a discrete group of isometries of hyperbolic space. The quotient ℍ3/Γ is hence a compact hyperbolic 3-orbifold, of which we study the hyperbolic volume and spectrum of closed geodesic lengths using SnapPea. One consequence is a volume estimate for a "hyperelliptic" manifold considered in [Mednykh and Vesnin 03].