A classification of R-Fuchsian subgroups of Picard modular groups

Abstract : 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).
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  • HAL Id : hal-01555728, version 1
  • ARXIV : 1707.00154

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Jouni Parkkonen, Frédéric Paulin. A classification of R-Fuchsian subgroups of Picard modular groups. avec quatre dessins. 2017. 〈hal-01555728〉

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