Detecting large groups

Let G be a finitely presented group, and let p be a prime. Then G is 'large' (respectively, 'p-large') if some normal subgroup with finite index (respectively, index a power of p) admits a non-abelian free quotient. This paper provides a variety of new methods for detecting whether G is large or p-large. These relate to the group's profinite and pro-p completions, to its first L2-Betti number and to the existence of certain finite index subgroups with 'rapid descent'. The paper draws on new topological and geometric techniques, together with a result on error-correcting codes.