# CONFLUENTES MATHEMATICI

A classification of $ℝ$-Fuchsian subgroups of Picard modular groups
Confluentes Mathematici, Tome 10 (2018) no. 2, pp. 75-92.

Given an imaginary quadratic extension $K$ of $ℚ$, we classify the maximal nonelementary subgroups of the Picard modular group $PU\left(1,2;{𝒪}_{K}\right)$ preserving a totally real totally geodesic plane in the complex hyperbolic plane ${ℍ}_{ℂ}^{2}$. We prove that these maximal $ℝ$-Fuchsian subgroups are arithmetic, and describe the quaternion algebras from which they arise. For instance, if the radius $\Delta$ of the corresponding $ℝ$-circle lies in $ℕ-\left\{0\right\}$, then the stabiliser arises from the quaternion algebra $\left(\phantom{\rule{-0.166667em}{0ex}}\frac{\Delta \phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}|{D}_{K}|}{ℚ}\phantom{\rule{-0.166667em}{0ex}}\right)$. We thus prove the existence of infinitely many orbits of $K$-arithmetic $ℝ$-circles in the hypersphere of ${ℙ}_{2}\left(ℂ\right)$.

Reçu le : 2018-12-01
Révisé le : 2018-09-02
Accepté le : 2018-12-02
Publié le : 2019-03-04
DOI : https://doi.org/10.5802/cml.51
Classification : 11F06,  11R52,  20H10,  20G20,  53C17,  53C55
Mots clés: Picard modular group, ball quotient, arithmetic Fuchsian groups, Heisenberg group, quaternion algebra, complex hyperbolic geometry, $ℝ$-circle, hypersphere
@article{CML_2018__10_2_75_0,
author = {Jouni Parkkonen and Fr\'ed\'eric Paulin},
title = {A classification of $\protect \mathbb{R}$-Fuchsian subgroups of Picard modular groups},
journal = {Confluentes Mathematici},
publisher = {Institut Camille Jordan},
volume = {10},
number = {2},
year = {2018},
pages = {75-92},
doi = {10.5802/cml.51},
language = {en},
url = {cml.centre-mersenne.org/item/CML_2018__10_2_75_0/}
}
Parkkonen, Jouni; Paulin, Frédéric. A classification of $\protect \mathbb{R}$-Fuchsian subgroups of Picard modular groups. Confluentes Mathematici, Tome 10 (2018) no. 2, pp. 75-92. doi : 10.5802/cml.51. https://cml.centre-mersenne.org/item/CML_2018__10_2_75_0/

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