Abstract
LetG 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 manyi; 2. the Cayley graphs ofG/G i (with respect to a fixed finite set of generators forG) form an expanding family; 3. infi(d(G i )−1)/[G:G i ]=0, whered(G i ) is the rank ofG i . The proof involves an analysis of the geometry and topology of finite Cayley graphs. Several applications of this result are given.
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