A classification of R-Fuchsian subgroups of Picard modular groups

Jouni Parkkonen (1) , Frédéric Paulin (2)

1 JYU - Department of Mathematics and Statistics [Jyväskylä Univ]
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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^2_C . We prove that these maximal R-Fuchsian subgroups are arithmetic, and describe the quaternion algebras from which they arise. For instance, if the radius ∆ of the corresponding R-circle lies in N, then the stabilizer arises from the quaternion algebra with Hilbert symbol (∆ , |D_K |) over Q. We thus prove the existence of infinitely many orbits of K-arithmetic R-circles in the hypersphere of P_2(C).

Mots clés: Picard group; ball quotient; arithmetic Fuchsian groups; Heisenberg group; quaternion algebra; complex hyperbolic geometry; R-circle; hypersphere

Domaines: Théorie des nombres [math.NT]; Géométrie différentielle [math.DG]; Théorie des groupes [math.GR]

Jouni Parkkonen, Frédéric Paulin. A classification of R-Fuchsian subgroups of Picard modular groups. 2017. ⟨hal-01555728⟩