Generators for the Euclidean Picard Modular Groups

TL;DR

This paper identifies explicit generators for Euclidean Picard modular groups over quadratic imaginary fields and analyzes their relations through fundamental domain combinatorics.

Contribution

It provides explicit generators and relations for these groups, advancing understanding of their algebraic and geometric structure.

Findings

Five explicit generators for Euclidean Picard modular groups are identified.

Relations of the isotropy subgroup are derived from fundamental domain analysis.

The structure of these groups is clarified through combinatorial methods.

Abstract

The goal of this article is to show that five explicitly given transformations, a rotation, two screw Heisenberg rotations, a vertical translation and an involution generate the Euclidean Picard modular groups with coefficient in the Euclidean ring of integers of a quadratic imaginary number field. We also obtain the relations of the isotropy subgroup by analysis of the combinatorics of the fundamental domain in Heisenberg group.