Growth and generation in SL2(Z/pZ)

We show that every subset of SL2(Z/pZ) grows rapidly when it acts on itself by the group operation. It follows readily that, for every set of generators A of SL2(Z/pZ), every element of SL2(Z/pZ) can be expressed as a product of at most O((logp)(c)) elements of A boolean OR A(-1), where c and the implied constant are absolute.