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