Torsion in 1-cusped Picard modular groups

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.