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 $\Gamma=\langle S\rangle$ 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 $S$.