Logarithmic bounds for the diameters of some Cayley graphs

Abstract

Let $S\subset {\text{GL}}_n(\mathbb{Z})$ be a finite symmetric set. We show that if the Zariski closure of $\Gamma =⟨S⟩$ is a product of ${\text{SL}}_d$ or a special affine linear group, then the diameter of the Cayley graph $\text{Cay}(\Gamma /\Gamma (q),{\pi }_q(S))$ is $O(\log q)$, where $q$ is an arbitrary positive integer, ${\pi }_q:\Gamma \to \Gamma /\Gamma (q)$ is the canonical projection induced by the reduction modulo $q$, and the implied constant depends only on $S$.

13 pages, part of the paper rewritten, to appear at Journal of Group Theory