A classification of ${\mathbb C}$-Fuchsian subgroups of Picard modular groups

Abstract : Given an imaginary quadratic extension $K$ of $\mathbb Q$, we give a classification of the maximal nonelementary subgroups of the Picard modular group $PSU_{1,2}({\mathcal O}_K)$ preserving a complex geodesic in the complex hyperbolic plane ${\mathbb H}^2_{\mathbb C}$. Complementing work of Holzapfel, Chinburg-Stover and M\"oller-Toledo, we show that these maximal ${\mathbb C}$-Fuchsian subgroups are arithmetic, arising from a quaternion algebra $\Big(\!\begin{array}{c} D\,,D_K\\\hline{\mathbb Q}\end{array} \!\Big)$ for some explicit $D\in{\mathbb N}-\{0\}$ and $D_K$ the discriminant of $K$. We thus prove the existence of infinitely many orbits of $K$-arithmetic chains in the hypersphere of ${\mathbb P}_2({\mathbb C})$.
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https://hal.archives-ouvertes.fr/hal-01205177
Contributor : Frédéric Paulin <>
Submitted on : Friday, September 25, 2015 - 9:00:46 AM
Last modification on : Thursday, January 11, 2018 - 6:27:13 AM

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  • HAL Id : hal-01205177, version 1
  • ARXIV : 1503.05801

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Jouni Parkkonen, Frédéric Paulin. A classification of ${\mathbb C}$-Fuchsian subgroups of Picard modular groups. 2015. 〈hal-01205177〉

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