A characterisation of large finitely presented groups

Abstract

In this paper, we will consider finitely presented groups that have a finite index subgroup which admits a surjective homomorphism onto a non-abelian free group. Gromov called these groups large [4]. Large groups have particularly nice properties (for example, super-exponential subgroup growth). They also play an important role in lowdimensional topology: it is a major conjecture that the fundamental group of any closed hyperbolic 3-manifold is large. Our main theorem is a characterisation of these groups in terms of the existence of a normal series where successive quotients are finite abelian groups with sufficiently large rank and order.