We prove that ifLLis a finite simple group of Lie type andAAa set of generators ofLL, then eitherAAgrows, i.e.,|A3|>|A|1+ε|A^3| > |A|^{1+\varepsilon }whereε\varepsilondepends only on the Lie rank ofLL, orA3=LA^3=L. This implies that for simple groups of Lie type of bounded rank a well-known conjecture of Babai holds, i.e., the diameter of any Cayley graph is polylogarithmic. We also obtain new families of expanders.A generalization of our proof yields the following. LetAAbe a finite subset ofSL(n,F)SL(n,\mathbb {F}),F\mathbb {F}an arbitrary field, satisfying|A3|≤K|A|\big |A^3\big |\le \mathcal {K}|A|. ThenAAcan be covered byKm\mathcal {K}^m, i.e., polynomially many, cosets of a virtually soluble subgroup ofSL(n,F)SL(n,\mathbb {F})which is normalized byAA, wheremmdepends onnn.