On the geometry of a Picard modular group

TL;DR

This paper investigates the geometric action of a specific Picard modular group on complex hyperbolic space, classifies its finite subgroups, and constructs a torsion-free subgroup with a detailed geometric understanding.

Contribution

It provides a detailed classification of finite subgroups and stabilizers in the Picard modular group and constructs an explicit torsion-free subgroup of finite index.

Findings

Classified conjugacy classes of maximal finite subgroups.

Described stabilizers of mirrors of complex reflections.

Constructed an explicit torsion-free subgroup of index 336.

Abstract

We study geometric properties of the action of the Picard modular group $\Gamma=PU(2,1,\mathcal{O}_7)$ on the complex hyperbolic plane $H^2_\mathbb{C}$, where $\mathcal{O}_7$ denotes the ring of algebraic integers in $\mathbb{Q}(i\sqrt{7})$. We list conjugacy classes of maximal finite subgroups in $\Gamma$ and give an explicit description of the Fuchsian subgroups that occur as stabilizers of mirrors of complex reflections in $\Gamma$. As an application, we describe an explicit torsion-free subgroup of index $336$ in $\Gamma$.