Expanders, rank and graphs of groups
Abstract
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/Gi (with respect to a fixed finite set of generators for G) form an expanding family; 3. infi (d(Gi)-1)/[G:Gi] = 0, where d(Gi) is the rank of Gi. 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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