# Expanders, rank and graphs of groups

- Received:

DOI: 10.1007/BF02773541

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

- 14 Citations
- 108 Views

## 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.