Growth in linear algebraic groups and permutation groups: towards a unified perspective

TL;DR

This paper explores the growth properties of finite simple groups, particularly linear algebraic groups and permutation groups, aiming to unify their understanding and extend product theorems to the alternating group Alt_n.

Contribution

It revisits the proof of diameter bounds for Alt_n, aligning it with methods used for linear algebraic groups, and establishes a new, albeit weaker, product theorem for Alt_n.

Findings

Revised proof of diameter bounds for Alt_n

Development of a partial product theorem for Alt_n

Enhanced understanding of growth phenomena in finite simple groups

Abstract

By now, we have a product theorem in every finite simple group $G$ of Lie type, with the strength of the bound depending only in the rank of $G$. Such theorems have numerous consequences: bounds on the diameters of Cayley graphs, spectral gaps, and so forth. For the alternating group Alt_n, we have a quasipolylogarithmic diameter bound (Helfgott-Seress 2014), but it does not rest on a product theorem. We shall revisit the proof of the bound for Alt_n, bringing it closer to the proof for linear algebraic groups, and making some common themes clearer. As a result, we will show how to prove a product theorem for Alt_n -- not of full strength, as that would be impossible, but strong enough to imply the diameter bound.