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

TL;DR

This work completes the arithmetic classification of Fuchsian subgroups of Picard modular groups by treating ${\mathbb R}$-Fuchsian subgroups. It shows that maximal nonelementary ${\mathbb R}$-Fuchsian subgroups of ${\Gamma_K}$ are arithmetic lattices arising from explicit quaternion algebras, and they correspond to stabilizers of rational points in the rational points set ${\mathbb P}{\rm AHI}({\mathbb Q})$. Each such subgroup is commensurable with a stabilizer of a point $[Y_\Delta]$, with the invariant $\Delta\in {\cal O}_K$ encoding its quaternion algebra via $(\frac{2\operatorname{Tr}\Delta,\ N(\Delta)|D_K|}{\mathbb Q})$ when $\operatorname{Tr}\Delta\neq0$, or reducing to ${\cal M}_2({\mathbb Q})$ otherwise; in particular there exist infinitely many $K$-arithmetic ${\mathbb R}$-circles. The paper builds a bridge between the geometry of ${\mathbb H}^2_{\mathbb C}$, the space of ${\mathbb R}$-circles, and the arithmetic of quaternion algebras and ternary quadratic forms, delivering a complete arithmetic description of all maximal ${\mathbb R}$-Fuchsian subgroups of the Picard groups. This advances understanding of the spectrum of real hyperbolic dynamics inside complex hyperbolic lattices and yields explicit constructors for dense orbits of $K$-arithmetic ${\mathbb R}$-circles in the boundary hypersphere.

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 $Δ$ of the corresponding $\mathbb R$-circle lies in $\mathbb N-\{0\}$, then the stabilizer arises from the quaternion algebra $\Big(\!\begin{array}{c} Δ\,,\, |D_K|\\\hline\mathbb Q\end{array} \!\Big)$. We thus prove the existence of infinitely many orbits of $K$-arithmetic $\mathbb R$-circles in the hypersphere of $\mathbb P_2(\mathbb C)$.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Remark 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Proposition 4.1
• proof