Expanders and growth of normal subsets in finite simple groups of Lie type

Abstract: We show that some classical results on expander graphs imply growth results on normal subsets in finite simple groups. As one application, it is shown that given a nontrivial normal subset $ A $ of a finite simple group $ G $ of Lie type of bounded rank, we either have $ G \setminus { 1 } \subseteq A^2 $ or $ |A^2| \geq |A|^{1+\epsilon} $, for $ \epsilon > 0 $. This improves a result of Gill, Pyber, Short and Szab\'o, and partially resolves a question of Pyber from the Kourovka notebook. We also propose a variant of Gowers' trick for two subsets, and give applications to products of large subsets in groups of Lie type, improving some results of Larsen, Shalev and Tiep.