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

Abstract

Given an imaginary quadratic extension $K$ of $\mathbb{Q}$, we classify the maximal nonelementary subgroups of the Picard modular group $\mathrm{PU}(1,2;{\mathcal{O}}_K)$ preserving a totally real totally geodesic plane in the complex hyperbolic plane ${\mathbb{H}}_{\mathbb{C}}^2$. 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 $\left(\begin{array}{c}\Delta ,|D_K|\\ \mathbb{Q}\end{array}\right)$. We thus prove the existence of infinitely many orbits of $K$-arithmetic $\mathbb{R}$-circles in the hypersphere of ${\mathbb{P}}_2(\mathbb{C})$.

This is an expanded version of an article to appear in Confluentes Math. This version contains an appendix with detailed computations for the proof of Proposition 5.3

References in corpus (1)

• Geodesic curves on Shimura surfaces

Cited by in corpus (1)

• Geodesic curves on Shimura surfaces