On the geometry of a Picard modular group

We study geometric properties of the action on the complex hyperbolic plane H-C(2) of the Picard modular group Gamma = PU(2, 1, O-7), where O-7 denotes the ring of algebraic integers in Q(i root 7). We list conjugacy classes of maximal finite subgroups in Gamma and give an explicit description of the Fuchsian subgroups that occur as stabilizers of mirrors of complex reflections in Gamma. As an application, we describe an explicit torsion-free subgroup of index 336 in Gamma.