Geodesic
A classification of ℝ-Fuchsian subgroups of Picard modular groups
Parkkonen, Jouni1 ; Paulin, Frédéric2
1 Department of Mathematics and Statistics, P.O. Box 35, 40014 University of Jyväskylä, Finland
2 Laboratoire de mathématique d’Orsay, UMR 8628 Univ. Paris-Sud et CNRS, Université Paris-Saclay, 91405 ORSAY Cedex, France
Confluentes Mathematici, Tome 10 (2018) no. 2, pp. 75-92 Cet article a éte moissonné depuis la source Numdam

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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 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).

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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

Parkkonen, Jouni 1 ; Paulin, Frédéric 2

1 Department of Mathematics and Statistics, P.O. Box 35, 40014 University of Jyväskylä, Finland
2 Laboratoire de mathématique d’Orsay, UMR 8628 Univ. Paris-Sud et CNRS, Université Paris-Saclay, 91405 ORSAY Cedex, France
Licence : CC-BY-NC-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {A classification of $\protect \mathbb{R}${-Fuchsian} subgroups of {Picard} modular groups},
     journal = {Confluentes Mathematici},
     pages = {75--92},
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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

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