Algorithms for Polycyclic-by-finite Groups Algorithms for Polycyclic-by-finite Groups Table of Contents

OF THE DISSERTATION Algorithms for Polycyclic-by-Finite Groups by Gretchen Ostheimer Dissertation Director: Professor Charles C. Sims Let R be a number eld. We present several algorithms for working with polycyclicbynite subgroups of GL(n;R). Let G be a subgroup of GL(n;R) given by a nite generating set of matrices. We describe an algorithm for deciding whether or not G is polycyclic-bynite. For polycyclic-bynite G, we describe an algorithm for deciding whether or not a given matrix is an element of G. We prove that an abstract group G has a faithful representation as a triangularizable subgroup of GL(n;Z) for some n if and only if G is polycyclic and the commutator subgroup of G is torsion-free nilpotent. Suppose G is a polycyclic group given by a consistent polycyclic presentation. We describe an algorithm for deciding whether or not G has a faithful representation as a triangularizable subgroup of GL(n;Z), as well as an algorithm for constructing such a representation if it exists. Preliminary experiments indicate that the algorithms described in this thesis are suitable for computer implementation. Further experimentation is needed to determine the range of input for which the algorithms are practical with current technology. ii Acknowledgements My sincere thanks go to my advisor, Charles Sims, for the immense help which he has given me over the last four years. Even when he was inundated with his other professional obligations, he always found time to meet with me and to give careful attention to my work. I have the deepest respect for him and the high standards which he sets for himself in his research, teaching and professional relationships, and I consider myself very lucky to have had this opportunity to work closely with him. I am very grateful to my o cemate, friend and colleague Eddie Lo. Often, an idle question unfolded into an afternoon of brainstorming about group theory, and without these conversations my research would not have been the same. He was there in good times as well as bad with a delightful mix of philosophical insight and humor. I would also like to thank John Shareshian for his consistent support as a friend and colleague. Many mathematicians and computer scientists helped me make the most of graduate school. David Johnson at Bell Labs has been a mentor for me throughout the last seven years. Richard Lyons helped see me through a particularly challenging time. Jerry Tunnell taught me to love number theory and helped me apply it in my research. John Dixon passed along a reference that allowed me to obtain a signi cant generalization. Ann Yasuhara helped me analyze the complexity of one of my bounds. I am grateful to the members of my committee for their careful reading and thoughtful feedback about this thesis. Finally, I'd like to thank my friends and family for their support throughout my graduate career. I'd like to thank my father for telling me I could do it when I was still young enough to believe everything that he said. And most of all, I'd like to thank my partner, Emmi Schatz, for her consistent love, encouragement and patience, not to mention all the fun we have had along the way. iii Dedication This thesis is dedicated to Emmi Schatz.