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

2024/06/18 by Saveliy V. Skresanov, Skresanov, Saveliy V.
Mathematics · #20D06 (Primary) 05C48 #20P05 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2406.12506

openalex publication_date 2024/06/18 · openalex created_date 2024/06/20 · openalex updated_date 2026/07/28

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 ∖ \ 1 \ ⊆ A² or |A²| ≥ |A|^1+ε , for ε> 0 . This improves a result of Gill, Pyber, Short and Szabó, 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.

Related

• Rapid expansion in finite simple groups
• Growth in finite simple groups of Lie type of bounded rank
• Growth in finite simple groups of Lie type
• Expansion in finite simple groups of Lie type
• Finite simple groups of Lie type as expanders