2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70383Let 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.13 pages; to appear in Israel J. MathGroup TheoryGeometric Topology20F65; 20E06, 20F05, 05C25text