Seminar

Given an imaginary quadratic number field K, we give a classification of the maximal nonelementary subgroups of the Picard modular group PSU_{1,2}(O_K) preserving a complex geodesic in the complex hyperbolic plane. Complementing work of Holzapfel, Chinburg-Stover and Möller-Toledo, we show that these maximal C-Fuchsian subgroups are arithmetic, arising from a quaternion algebra over Q with Hilbert symbol (D,D_K) for some explicit positive integer D and with D_K the discriminant of K. We thus prove the existence of infinitely many orbits of K-arithmetic Cartan chains in Poincaré's hypersphere in the complex projective plane. This is a joint work with Jouni Parkkonen.