Torsion in 1-cusped Picard modular groups

TL;DR

This paper introduces an effective method for constructing fundamental domains of Picard modular groups, enabling classification of torsion elements, group presentations, and subgroup construction for certain discriminants.

Contribution

The paper develops a systematic computational approach to analyze Picard modular groups with class number one, providing new classifications and explicit group structures.

Findings

Classified conjugacy classes of torsion elements

Derived short presentations for the groups

Constructed neat subgroups of small index

Abstract

We present a systematic effective method to construct coarse fundamental domains for the action of the Picard modular groups $PU(2,1,\mathcal{O}_d)$ where $\mathcal{O}_d$ has class number one, i.e. $d=1,2,3,7,11,19,43,67,163$. The computations can be performed quickly up to the value $d=19$. As an application of this method, we classify conjugacy classes of torsion elements, deduce short presentations for the groups, and construct neat subgroups of small index.