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We prove that if L is a finite simple group of Lie type and A a set of generators of L, then A grows i.e |A 3 | > |A| 1+ε where ε depends only on the Lie rank of L, or A 3 = L. This implies that for a family of simple groups L of Lie type the diameter of any Cayley graph is polylogarithmic in |L|. 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 | ≤ K|S| for some K. Then G has two normal subgroups H ≥ 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.

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Abstract. We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If k is the minimal degree of a representation of the finite group G, then for every subset B of G with |B| > |G|/k 1/3 we have B 3 = G. We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs.On the other hand, we prove a version of Jordan's theorem which implies that if k ≥ 2, then G has a proper subgroup of index at most c 0 k 2 for some constant c 0 , hence a product-free subset of size at least |G|/(ck). This answers a question of Gowers.