Presentations for the Euclidean Picard modular groups

TL;DR

This paper applies a geometric method to derive presentations for certain Euclidean Picard modular groups, completing the classification for all such groups with Euclidean domain entries.

Contribution

It extends Mark and Paupert's method to specific Picard modular groups with Euclidean domains, providing explicit group presentations.

Findings

Presentations for Picard modular groups with d=2 and d=11.

Completes the list of Euclidean Picard modular groups.

Advances understanding of arithmetic lattices in complex hyperbolic space.

Abstract

Mark and Paupert devised a general method for obtaining presentations for arithmetic non-cocompact lattices, $\Gamma$, in isometry groups of negatively curved symmetric spaces. The method involves a classical theorem of Macbeath applied to a $\Gamma$-invariant covering by horoballs of the negatively curved symmetric space upon which $\Gamma$ acts. In this paper, we will discuss the application of their method to the Picard modular groups, $\textrm{PU}(2,1;\mathcal{O}_{d})$, when $d=2,11$, and obtain presentations for these groups, which completes the list of presentations for Picard modular groups whose entries lie in Euclidean domains, namely those with $d=1,2,3,7,11$.