Improved computation of fundamental domains for arithmetic Fuchsian groups

TL;DR

This paper presents an improved, more efficient algorithm for computing fundamental domains of arithmetic Fuchsian groups, implemented in PARI/GP, with demonstrated performance gains over previous Magma-based methods.

Contribution

It combines and enhances existing algorithms to produce a faster method for fundamental domain computation of arithmetic Fuchsian groups.

Findings

Significant reduction in running times compared to previous implementations

Successful implementation in PARI/GP demonstrating practical efficiency

Enhanced algorithm applicable to a broader class of groups

Abstract

A practical algorithm to compute the fundamental domain of an arithmetic Fuchsian group was given by Voight, and implemented in Magma. It was later expanded by Page to the case of arithmetic Kleinian groups. We combine and improve on parts of both algorithms to produce a more efficient algorithm for arithmetic Fuchsian groups. This algorithm is implemented in PARI/GP, and we demonstrate the improvements by comparing running times versus the live Magma implementation.