  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|^(1+epsilon) where epsilon
depends only on the Lie rank of L, or AAA=L. This implies that for a family of
simple groups L of Lie type the diameter of any Cayley graph is polylogarithmic
in |L|. Combining our result on growth with known results of Bourgain,Gamburd
and Varj\'u it follows that if LAMBDA is a Zariski-dense subgroup of SL(d,Z)
generated by a finite symmetric set S, then for square-free moduli m which are
relatively prime to some number m_0 the Cayley graphs Gamma(SL(d,m),pi_m(S))
form an expander family.
