Israel Journal of Mathematics

, Volume 146, Issue 1, pp 357–370

Expanders, rank and graphs of groups


DOI: 10.1007/BF02773541

Cite this article as:
Lackenby, M. Isr. J. Math. (2005) 146: 357. doi:10.1007/BF02773541


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

Copyright information

© The Hebrew University Magnes Press 2005

Authors and Affiliations

  1. 1.Mathematical InstituteOxford UniversityOxfordUK