Presentations for the Euclidean Picard modular groups

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 $d=1,2,3,7,11$.