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Jouni Parkkonen; Frédéric Paulin
A classification of $\protect \mathbb{R}$-Fuchsian subgroups of Picard modular groups
Confluentes Mathematici, 10 no. 2 (2018), p. 75-92, doi: 10.5802/cml.51
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Class. Math.: 11F06, 11R52, 20H10, 20G20, 53C17, 53C55
Keywords: Picard modular group, ball quotient, arithmetic Fuchsian groups, Heisenberg group, quaternion algebra, complex hyperbolic geometry, $\protect \mathbb{R}$-circle, hypersphere

Résumé - 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}-\lbrace 0\rbrace $, then the stabiliser arises from the quaternion algebra $\Big (\!\frac{\Delta \,,\, |D_K|}{\mathbb{Q}}\!\Big )$. We thus prove the existence of infinitely many orbits of $K$-arithmetic $\mathbb{R}$-circles in the hypersphere of $\mathbb{P}_2(\mathbb{C})$.

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