Proving Finitely Presented Groups are Large by Computer

Abstract

We present a theoretical algorithm that given any finite presentation of a group as input, will terminate with answer yes if and only if the group is large. We then implement a practical version of this algorithm using MAGMA and apply it to a range of presentations. Our main focus is on two-generator one-relator presentations, for which we have a complete picture of largeness if the relator has exponent sum zero in one generator and word length at most 12, as well as if the relator is in the commutator subgroup and has word length at most 18. Indeed, all but a tiny number of presentations define large groups. Finally, we look at fundamental groups of closed hyperbolic 3-manifolds, for which the algorithm readily determines that at least a quarter of the groups in the SnapPea closed census are large.