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

TL;DR

This paper classifies maximal nonelementary complex Fuchsian subgroups of Picard modular groups over imaginary quadratic fields, showing they are arithmetic and related to specific quaternion algebras, revealing infinitely many orbits of K-arithmetic chains.

Contribution

It provides a complete classification of maximal C-Fuchsian subgroups of Picard modular groups, demonstrating their arithmetic nature and connection to quaternion algebras, extending previous work.

Findings

Maximal C-Fuchsian subgroups are arithmetic.

Existence of infinitely many K-arithmetic chains.

Classification involves explicit quaternion algebras.

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 $\mathbb P_2(\mathbb C)$.