Growth in $\mathrm{SL}_3$(ℤ/$p$ℤ)

Abstract

Let $G = \mathrm{SL}_3$(ℤ/$p$ℤ), $p$ a prime. Let A be a set of generators of G. Then A grows under the group operation.

To be precise: denote by |S| the number of elements of a finite set S. Assume |A| < |G|1-ε for some ε > 0. Then |A ∙ A ∙ A| > |A|1+δ, where δ > 0 depends only on ε.