Growth and generation in SL2(ℤ/pℤ)

Abstract

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((log p)c) elements of A ∪ A-1, where c and the implied constant are absolute.