Residual Deficiency

We introduce a new real valued invariant for finitely presented groups called residual deficiency. Its main property is the following. Let G be a finitely presented group. If the residual deficiency of G is greater than one, then G has a finite index subgroup with deficiency greater than one. The latter property is strong. For instance, such a group is large, has positive rank gradient and positive first L^2-Betti number, among other properties. We also compute the residual deficiency of some well known families of presentations, prove that residual deficiency minus one is supermultiplicative with respect to finite index normal subgroups and find lower bounds for the residual deficiency of quotients.