Presentations for Singly-Cusped Bianchi Groups

TL;DR

This paper provides explicit group presentations for singly-cusped Bianchi groups, expanding known cases to include those with discriminants -43, -67, and -163, using geometric methods involving horoballs.

Contribution

It offers the first explicit presentations for certain Bianchi groups with specific discriminants, utilizing a geometric approach based on Macbeath's theorem.

Findings

Complete list of presentations for specified Bianchi groups

New explicit presentations for groups with d=-43, -67, -163

Application of Macbeath's theorem to hyperbolic geometry

Abstract

We produce a complete list of group presentations for singly-cusped Bianchi groups, $PSL_2 (\mathcal{O}_d)$ where $\mathcal{O}_d$ is the ring of integers for $\mathbb{Q}(\sqrt{d})$ and $d$ is -1, -2, -3, -7, -11, -19, -43, -67, or -163. To do this, we apply a theorem due to Macbeath to a sufficiently large horoball, treating the Bianchi groups as discrete subgroups of isometries for $\mathbf{H}^3$. As far as we know, explicit presentations were not previously known when $d$ is -43, -67, or -163.