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 $\Gamma$.
Categories: math.GT math.AG