A classification of C-Fuchsian subgroups of Picard modular groups - Parkkonen, Jouni; Paulin, Frédéric (MATHEMATICA SCANDINAVICA, 2017)
Given an imaginary quadratic extension K of Q, 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 H^2_C. 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 (\!arrayc D D_K Qarray \!) for some explicit DN-{0} and D_K the discriminant of K. We thus prove the existence of infinitely many orbits of K-arithmetic chains in the hypersphere of P_2(C).