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 are non-obtuse, Andreev's Theorem 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 proof of Andreev's Theorem as well as a convenient way to construct three-dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles. As an application, we construct compact hyperbolic polyhedra having dihedral angles that are (proper) integer sub-multiples of $\pi$, so that the group $\Gamma$ generated by reflections in the faces is a discrete group of isometries of hyperbolic space. The quotient ${\mathbb{H}}^3/\Gamma$ 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 by Mednykh and Vesnin (see references).

Revision includes an experimental study of volumes and geodesic length spectra for certain compact polyhedral orbifolds. Many additional references added

References in corpus (1)

• Numerical Study of Length Spectra and Low-lying Eigenvalue Spectra of Compact Hyperbolic 3-manifolds