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

2015/03/19 by Jouni Parkkonen, Parkkonen, Jouni, Frédéric Paulin +1
Mathematics · #11E39 #11F06 #11N45 #20G20 #53C17 #53C22 #53C55 #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #math.GR #math.NT #msc:11E39 #msc:11F06 #msc:11N45 #msc:20G20 #msc:53C17 #msc:53C22 #msc:53C55

paper · pdf · doi:10.48550/arxiv.1503.05801

Added references and attribution to earlier work

arxiv created 2015/04/08 · arxiv updated 2015/04/09

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 H²_\mathbb C. Complementing work of Holzapfel, Chinburg-Stover and Möller-Toledo, we show that these maximal \mathbb C-Fuchsian subgroups are arithmetic, arising from a quaternion algebra ( D ,D_K
\mathbb QQ ) for some explicit D∈\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 P₂(\mathbb C).

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