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

TL;DR

This paper connects expander graph theory with growth in finite simple groups of Lie type, showing that normal subsets either cover almost the entire group or grow significantly when squared.

Contribution

It introduces new growth results for normal subsets in finite simple groups of Lie type, improving previous bounds and partially answering open questions.

Findings

Normal subsets either cover all but the identity or grow substantially when squared.

Established a variant of Gowers' trick applicable to two subsets.

Improved bounds on product growth of large subsets in 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.