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Mark and Paupert concocted a general method for producing presentations for arithmetic non-cocompact lattices, \(\Gamma\), in isometry groups of negatively curved symmetric spaces. To get around the difficulty of constructing fundamental domains in spaces of variable curvature, their method invokes a classical theorem of Macbeath applied to a \(\Gamma\)-invariant covering by horoballs of the negatively curved symmetric space upon which \(\Gamma\) acts. This thesis aims to explore the application of their method to the Picard modular groups, PU\((2,1;\mathcal{O}_{d})\), acting on \(\mathbb{H}_{\C}^2\). This document contains the derivations for the group presentations corresponding to \(d=2,11\), which completes the list of presentations for Picard modular groups whose entries lie in Euclidean domains, namely those with \(d=1,2,3,7,11\). There are differences in the method's application when the lattice of interest has multiple cusps. \(d = 5\) is the smallest value of \(d\) for which the corresponding Picard modular group, \(\PU(2,1;\mathcal{O}_5)\), has multiple cusps, and the method variations become apparent when working in this case.
ContributorsPolletta, David Michael (Author) / Paupert, Julien H (Thesis advisor) / Kotschwar, Brett (Committee member) / Fishel, Susanna (Committee member) / Kawski, Matthias (Committee member) / Childress, Nancy (Committee member) / Arizona State University (Publisher)
Created2021