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161554 https://hdl.handle.net/2286/R.2.N.161554 http://rightsstatements.org/vocab/InC/1.0/ All Rights Reserved 2021 143 pages Doctoral Dissertation Academic theses eng Polletta, David Michael Paupert, Julien H Kotschwar, Brett Fishel, Susanna Kawski, Matthias Childress, Nancy Arizona State University Partial requirement for: Ph.D., Arizona State University, 2021 Field of study: Mathematics 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. Mathematics Algebra Complex Hyperbolic Geometry Geometric Group Theory Geometry Number Theory Semisimple Lie groups The Geometry of 1-cusped and 2-cusped Picard Modular Groups