On Diameters of Cayley Graphs over Matrix Groups

Abstract

We establish for the matrix group $G={\mathrm{SL}}_n\left({\mathbb{F}}_p\right)$ that there exist absolute constants $c\in \left(0,1\right)$ and $C>0$ such that any symmetric generating set $A$, with $\left|A\right|\ge {\left|G\right|}^{1-c}$ has a covering number $\le C{n}^2.$ This result is sharp up to the value of the constant $C>0$.