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

TL;DR

This paper classifies maximal nonelementary real Fuchsian subgroups of Picard modular groups, showing they are arithmetic and describing their quaternion algebra origins, revealing infinitely many orbits of K-arithmetic R-circles.

Contribution

It provides a complete classification of these subgroups, demonstrating their arithmetic nature and explicitly linking them to quaternion algebras, which was previously unknown.

Findings

Maximal R-Fuchsian subgroups are arithmetic.

Explicit description of quaternion algebras for these subgroups.

Existence of infinitely many orbits of K-arithmetic R-circles.

Abstract

Given an imaginary quadratic extension $K$ of $\mathbb Q$, we classify the maximal nonelementary subgroups of the Picard modular group $\operatorname{PU}(1,2;\mathcal O_K)$ preserving a totally real totally geodesic plane in the complex hyperbolic plane $\mathbb H^2_\mathbb C$. We prove that these maximal $\mathbb R$-Fuchsian subgroups are arithmetic, and describe the quaternion algebras from which they arise. For instance, if the radius $\Delta$ of the corresponding $\mathbb R$-circle lies in $\mathbb N-\{0\}$, then the stabilizer arises from the quaternion algebra $\Big(\!\begin{array}{c} \Delta\,,\, |D_K|\\\hline\mathbb Q\end{array} \!\Big)$. We thus prove the existence of infinitely many orbits of $K$-arithmetic $\mathbb R$-circles in the hypersphere of $\mathbb P_2(\mathbb C)$.