On the geometry of a Picard modular group

We study geometric properties of the action on the complex hyperbolic plane H^2_\mathbb{C} of the Picard modular group \Gamma=\mathrm{PU}(2,1,\mathcal{O}_7) , where \mathcal{O}_7 denotes the ring of algebraic integers in \mathbb{Q}(i\sqrt{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 .