On the geometry of a Picard modular group

Abstract

We study geometric properties of the action on the complex hyperbolic plane $H_{\mathbb{C}}^2$ 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 $\Gamma$.