Expanders, rank and graphs of groups

2004/03/08 by Marc Lackenby, Lackenby, Marc · 1 citation
Mathematics · #05C25 #20E06 #20F05 #20F65 #Advanced Operator Algebra Research #FOS: Mathematics #Finite Group Theory Research #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #math.GR #math.GT #msc:05C25 #msc:20E06 #msc:20F05 #msc:20F65

paper · pdf · doi:10.48550/arxiv.math/0403127

13 pages; to appear in Israel J. Math

openalex publication_date 2004/03/08 · arxiv created 2005/04/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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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