Growth and generation in $\mathrm{SL}_2(\mathbb{Z}/p \mathbb{Z})$

Abstract

We show that every subset of $\mathrm{SL}_2(\mathbb{Z}/p\mathbb{Z})$ grows rapidly when it acts on itself by the group operation. It follows readily that, for every set of generators $A$ of $\mathrm{SL}_2(\mathbb{Z}/p\mathbb{Z})$, every element of $\mathrm{SL}_2(\mathbb{Z}/p\mathbb{Z})$ can be expressed as a product of at most $O((\log p)^c)$ elements of $A \cup A^{-1}$, where $c$ and the implied constant are absolute.