A classification of $\mathbb{C}$-Fuchsian subgroups of Picard modular groups
Keyword(s):
Given an imaginary quadratic extension $K$ of $\mathbb{Q}$, we give a classification of the maximal nonelementary subgroups of the Picard modular group $\operatorname{PSU}_{1,2}(\mathcal{O}_K)$ 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 $\mathbb{P}_2(\mathbb{C})$.
2005 ◽
Vol 55
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pp. 399-439
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2013 ◽
Vol 55
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pp. 645-654
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2007 ◽
Vol 135
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pp. 3349-3358
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2006 ◽
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pp. 1151-1173
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1989 ◽
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pp. 350-367
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Vol 91
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pp. 421-429
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2020 ◽
Vol 44
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pp. 1469-1475
2012 ◽
Vol 08
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pp. 983-992
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