On the geometry of a Picard modular group

Abstract

We study geometric properties of the action of the Picard modular group $Γ=PU(2,1,\mathcal{O}_7)$ 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 $Γ$ and give an explicit description of the Fuchsian subgroups that occur as stabilizers of mirrors of complex reflections in $Γ$. As an application, we describe an explicit torsion-free subgroup of index $336$ in $Γ$.