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.