  Let G be a finitely presented group, and let {G_i} 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. G_i 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.
