On the geometry of a Picard modular group

2021/07/21 by Martin Deraux, Deraux, Martin
Mathematics · #Geometric and Algebraic Topology #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology

paper · doi:10.48550/arxiv.2107.09969

Abstract

We study geometric properties of the action of the Picard modular group Γ=PU(2,1,O₇) on the complex hyperbolic plane H²_ℂ, where O₇ denotes the ring of algebraic integers in ℚ(i√(7)). We list conjugacy classes of maximal finite subgroups in Γ and give an explicit description of the Fuchsian subgroups that occur as stabilizers of mirrors of complex reflections in Γ. As an application, we describe an explicit torsion-free subgroup of index 336 in Γ.

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