  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 of bounded rank the diameter of any Cayley graph is
polylogarithmic in |L|. We obtain a similar bound for the diameters of all
Cayley graphs of perfect subgroups of GL(n,p) generated by their elements of
order p. We also obtain some new families of expanders. We also prove the
following partial extension. Let G be a subgroup of GL(n,p), p a prime, and S a
symmetric set of generators of G satisfying |S^3|\le K|S| for some K. Then G
has two normal subgroups H\ge P such that H/P is soluble, P is contained in S^6
and S is covered by K^c cosets of H where c depends on n. We obtain results of
similar flavour for sets generating infinite subgroups of GL(n,F), F an
arbitrary field.
