New computational methods for analysing finitely-presented groups

Abstract

In this thesis, we describe three new and/or improved methods for the analysis of finitely-presented groups and show their usefulness in a variety of contexts. The first procedure creates a labelled coset graph which can be used for rewriting and also for finding expressions for subgroup elements in terms of given generators. We use this procedure to find nice generating sets for torsion-free subgroups of finite index in ordinary triangle groups, with implications for the study of regular maps and automorphism groups of compact Riemann surfaces. Our second procedure is an improvement of the package PEACE by Havas and Ramsay. This package uses coset enumeration to find a proof for subgroup inclusion, by way of a proof word (which is a sequence of the elements of the supergroup and various brackets indicating two methods of simplification, the equality of which proves the inclusion). We use this procedure to find new generating sets and presentations for the special linear group SL(3, Z). Finally, we give a version of the low-index subgroups algorithm with added capabilities for finding specific types of subgroups by way of avoiding the inclusion of specified words. We use this algorithm to find torsion-free subgroups of Coxeter groups, with implications for the construction of hyperbolic manifolds.