Expanders, rank and graphs of groups

Let G be a finitely presented group, and let {Gi} be a collection of finite index normal subgroups that is closed under intersections. Then, we prove that at least one of the following must hold: 1. Gi is an amalgamated free product or HNN extension, for infinitely many i; 2. the Cayley graphs of G/G(i) (with respect to a fixed finite set of generators for G) form an expanding family; 3. inf(i)(d(G(i)) - 1)/[G : G(i)] = 0, where d(G(i)) is the rank of G(i). The proof involves an analysis of the geometry and topology of finite Cayley graphs. Several applications of this result are given.