Expanders, rank and graphs of groups

Details Expanders, rank and graphs of groups Expanders, rank and graphs of groups

2004-03-08

arxiv.org/abs/math/0403127v2

Mathematics

Group Theory

13 pages; to appear in Israel J. Math

Scientific paper

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.

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