On the geometry of a Picard modular group

Abstract: We study geometric properties of the action of the Picard modular group $\Gamma=PU(2,1,\mathcal{O}7)$ on the complex hyperbolic plane $H^2\mathbb{C}$, 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 $\Gamma$.