Growth in finite simple groups of Lie type of bounded rank

TL;DR

This paper proves that finite simple groups of Lie type of bounded rank exhibit rapid growth in their generating sets, leading to polylogarithmic diameters of Cayley graphs and new expander families.

Contribution

It establishes growth properties for simple groups of Lie type of bounded rank and derives diameter bounds and expander constructions, extending to subgroups of GL(n,p).

Findings

A symmetric generating set A in such groups satisfies |AAA| > |A|^{1+epsilon}.

Cayley graphs of these groups have polylogarithmic diameter.

New families of expander graphs are constructed.

Abstract

We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A|^{1+epsilon} where epsilon depends only on the Lie rank of L, or AAA=L. This implies that for a family of simple groups L of Lie type of bounded rank the diameter of any Cayley graph is polylogarithmic in |L|. We obtain a similar bound for the diameters of all Cayley graphs of perfect subgroups of GL(n,p) generated by their elements of order p. We also obtain some new families of expanders. We also prove the following partial extension. Let G be a subgroup of GL(n,p), p a prime, and S a symmetric set of generators of G satisfying |S^3|\le K|S| for some K. Then G has two normal subgroups H\ge P such that H/P is soluble, P is contained in S^6 and S is covered by K^c cosets of H where c depends on n. We obtain results of similar flavour for sets generating infinite subgroups of GL(n,F), F an arbitrary field.