Growth and generation in SL_2(Z/pZ)

Abstract

We show that every subset of SL_2(Z/pZ) grows rapidly when it acts on itself by the group operation. It follows readily that, for every set of generators A of SL_2(Z/pZ), every element of SL_2(Z/pZ) can be expressed as a product of at most O((log p)^c) elements of the union of A and A^{-1}, where c and the implied constant are absolute.

21 pages; fourth version - lemmas on escape amended and improved; to appear in Annals of Mathematics

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