Proving finitely presented groups are large by computer

TL;DR

This paper introduces a theoretical and practical algorithm to determine if finitely presented groups are large, successfully applying it to various classes including 3-manifold groups, revealing most are large.

Contribution

The paper develops a new algorithm for identifying large groups from finite presentations and implements it in Magma, providing comprehensive results for specific classes of groups.

Findings

Most 2-generator 1-relator groups with certain properties are large.

The algorithm determines that about 25% of groups in the Snappea census are large.

Theoretical results cover broad classes of groups, including hyperbolic 3-manifold groups.

Abstract

We present a theoretical algorithm which, 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 2-generator 1-relator presentations where 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 when 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, where the algorithm readily determines that a quarter of the groups in the Snappea closed census are large.