Computing a Dirichlet domain for a hyperbolic surface

TL;DR

This paper presents an algorithm that computes explicit Dirichlet domains for closed hyperbolic surfaces using geometric and topological data, with proven polynomial time complexity.

Contribution

The paper introduces a novel algorithm that efficiently computes Dirichlet domains for hyperbolic surfaces, combining geometric and topological data structures.

Findings

Algorithm terminates in polynomial time

Uses fundamental polygons with side pairings as input

Effectively integrates geometry and topology

Abstract

The goal of this paper is to exhibit and analyze an algorithm that takes a given closed orientable hyperbolic surface and outputs an explicit Dirichlet domain. The input is a fundamental polygon with side pairings. While grounded in topological considerations, the algorithm makes key use of the geometry of the surface. We introduce data structures that reflect this interplay between geometry and topology and show that the algorithm finishes in polynomial time, in terms of the initial perimeter and the genus of the surface.