Expansion in perfect groups
Group Theory
2013-01-28 v3 Combinatorics
Number Theory
Abstract
Let Ga be a subgroup of GL_d(Q) generated by a finite symmetric set S. For an integer q, denote by Ga_q the subgroup of Ga consisting of the elements that project to the unit element mod q. We prove that the Cayley graphs of Ga/Ga_q with respect to the generating set S form a family of expanders when q ranges over square-free integers with large prime divisors if and only if the connected component of the Zariski-closure of Ga is perfect.
Cite
@article{arxiv.1108.4900,
title = {Expansion in perfect groups},
author = {Alireza Salehi Golsefidy and Péter P. Varjú},
journal= {arXiv preprint arXiv:1108.4900},
year = {2013}
}
Comments
62 pages, no figures, revision based on referee's comments: new ideas are explained in more details in the introduction, typos corrected, results and proofs unchanged