Resumen de A classification of C -Fuchsian subgroups of Picard modular groups

Jouni Parkkonen, Frédéric Paulin

• Given an imaginary quadratic extension K of Q, we give a classification of the maximal nonelementary subgroups of the Picard modular group PSU1,2(OK) preserving a complex geodesic in the complex hyperbolic plane H2C. Complementing work of Holzapfel, Chinburg-Stover and M\"oller-Toledo, we show that these maximal C-Fuchsian subgroups are arithmetic, arising from a quaternion algebra (D,DKQ) for some explicit D∈N−{0} and DK the discriminant of K. We thus prove the existence of infinitely many orbits of K-arithmetic chains in the hypersphere of P2(C)