# Marston Conder: Finding subgroups of interest in finitely-presented groups

Abstract: This talk will describe some methods for finding particular kinds of subgroups of finite index in finitely-presented groups, such as those which are torsion-free or are generated by a small number of elements. Such methods include the use of the standard ‘Low index subgroups’ algorithm (due to Sims), adaptations of that to trim the search tree, taking intersections of subgroups found, use of the Reidemeister-Schreier algorithm to dig deeper into small-index subgroups found, and a permutational approach using regular orbits of a known finite subgroup. Applications will be given in the case of certain finitely-presented groups, including one from almost 20 years ago involving the [ 5, 3, 3, 3 ] Coxeter group to find the compact hyperbolic 4-manifold of smallest known volume, and more recent examples that complete some work by Milnor (in the 1970s) and Lorimer (in the 1990s) on determining small-volume hyperbolic 3-manifolds with particular parameters, and others that show (somewhat surprisingly) that an infinite hyperbolic group of rank up to 7 can sometimes contain a 2-generator subgroup of small finite index.