Generators of the Gauss-Picard modular groups in three complex dimensions

TL;DR

This paper proves that the Gauss-Picard modular group in three complex dimensions can be generated by five specific transformations, providing a potential method applicable to other higher-dimensional Euclidean-Picard groups.

Contribution

It identifies a finite generating set for the three-dimensional Gauss-Picard modular group, extending the understanding of its structure and potential generalization to higher dimensions.

Findings

The group is generated by five transformations.

The method may extend to other higher-dimensional groups.

Provides explicit generators for PU(3,1;Z[i])

Abstract

In this paper, we prove that the Gauss--Picard modular group PU(3,1;Z[i])in three complex dimensions can be generated by five given transformations: two Heisenberg translations, two Heisenberg rotations and an involution. Indeed, our method may work for the other higher dimensional Euclidean--Picard modular groups.