Tiling Hyperbolic Manifolds: Algorithms and Applications

TL;DR

This paper presents a novel algorithm for tiling hyperbolic 3-manifolds, enabling computation of cusp structures and decompositions with verified accuracy, advancing geometric understanding and computational methods in hyperbolic geometry.

Contribution

The paper introduces a new tiling algorithm for hyperbolic 3-manifolds, computes cusp area matrices, and provides verified interval methods for accurate geometric calculations.

Findings

Computed maximal cusp area matrices for hyperbolic 3-manifolds

Successfully determined Epstein-Penner decompositions

Provided simplified formulas for hyperboloid model distances

Abstract

We introduce a new tiling algorithm for hyperbolic 3-manifolds. We use it to compute the maximal cusp area matrix; this completely characterizes the space of all embedded and disjoint cusp neighborhoods. As another application of our work, we find the Epstein-Penner decomposition answering a challenge of Sakuma and Weeks. We furthermore provide the refinements needed to make our algorithm verified: producing intervals provably containing the correct answer. As key ingredient for our work and perhaps of independent interest, we give new and simpler expressions for the distances between points, lines, and planes in the hyperboloid model.