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.