A classification of $\mathbb C$-Fuchsian subgroups of Picard modular groups

Abstract

Given an imaginary quadratic extension $K$ of $\mathbb Q$, we give a classification of the maximal nonelementary subgroups of the Picard modular group $\operatorname{PSU}_{1,2}(\mathcal O_K)$ preserving a complex geodesic in the complex hyperbolic plane $\mathbb H^2_\mathbb C$. Complementing work of Holzapfel, Chinburg-Stover and M\"oller-Toledo, we show that these maximal $\mathbb C$-Fuchsian subgroups are arithmetic, arising from a quaternion algebra $\Big(\!\begin{array}{c} D\,,D_K\\\hline\mathbb QQ\end{array} \!\Big)$ for some explicit $D\in\mathbb N-\{0\}$ and $D_K$ the discriminant of $K$. We thus prove the existence of infinitely many orbits of $K$-arithmetic chains in the hypersphere of $\mathbb P_2(\mathbb C)$.