Growth in Finite Simple Groups of Lie Type

We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A| where ε depends only on the Lie rank of L, or AAA = L. This implies that for a family of simple groups L of Lie type of bounded rank the diameter of any Cayley graph is polylogarithmic in |L|. Combining our result on growth with known results of Bourgain, Gamburd and Varjú it follows that if Λ is a Zariski-dense subgroup of SL(d,Z) generated by a finite symmetric set S, then for squarefree moduli m, which are relatively prime to some number m0, the Cayley graphs Γ(SL(d,Z/mZ), πm(S)) form an expander family.