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

Jouni Parkkonen; Frédéric Paulin

doi:10.5802/cml.51

A classification of

ℝ

-Fuchsian subgroups of Picard modular groups

Jouni Parkkonen ¹; Frédéric Paulin ²

¹ Department of Mathematics and Statistics, P.O. Box 35, 40014 University of Jyväskylä, Finland
² Laboratoire de mathÃ©matique d’Orsay, UMR 8628 Univ. Paris-Sud et CNRS, Université Paris-Saclay, 91405 ORSAY Cedex, France

Confluentes Mathematici, Volume 10 (2018) no. 2, pp. 75-92.

Abstract

Given an imaginary quadratic extension $K$ of $ℚ$ , we classify the maximal nonelementary subgroups of the Picard modular group $PU (1, 2; 𝒪_{K})$ 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 $Δ$ of the corresponding $ℝ$ -circle lies in $ℕ - {0}$ , then the stabiliser arises from the quaternion algebra $(\frac{Δ, | D_{K} |}{ℚ})$ . We thus prove the existence of infinitely many orbits of $K$ -arithmetic $ℝ$ -circles in the hypersphere of $ℙ_{2} (ℂ)$ .

Received: 2018-11-30
Revised: 2018-09-01
Accepted: 2018-12-01
Published online: 2019-03-03

MR: 3928225

DOI: 10.5802/cml.51

Classification: 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

Author's affiliations:

Jouni Parkkonen ¹; Frédéric Paulin ²

License:

CC-BY-NC-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Jouni Parkkonen and Fr\'ed\'eric Paulin},
     title = {A classification of $\protect \mathbb{R}${-Fuchsian} subgroups of {Picard} modular groups},
     journal = {Confluentes Mathematici},
     pages = {75--92},
     publisher = {Institut Camille Jordan},
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Jouni Parkkonen; Frédéric Paulin. A classification of $\protect \mathbb{R}$-Fuchsian subgroups of Picard modular groups. Confluentes Mathematici, Volume 10 (2018) no. 2, pp. 75-92. doi : 10.5802/cml.51. https://cml.centre-mersenne.org/articles/10.5802/cml.51/

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