A characterisation of large finitely presented groups
  A group is known as `large' if some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. In this paper, we give a necessary and sufficient condition for a finitely presented group to be large, in terms of the existence of a normal series where successive quotients are finite abelian groups with sufficiently large rank and order. The proof of this result involves an analysis of the geometry and topology of finite Cayley graphs. Theorems of Baumslag and Pride, and their extensions by Gromov and Stohr, on groups with more generators than relations, follow immediately. 
Marc Lackenby