Growth in SL_3(Z/pZ)
  Let G=SL_3(Z/pZ), 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-\epsilon} for some \epsilon>0. Then |A\cdot A\cdot A|>|A|^{1+\delta}, where \delta>0 depends only on \epsilon.   We also study subsets A\subset G that do not generate G. Other results on growth and generation follow. 
H. A. Helfgott