# An improved diameter bound for finite simple groups of Lie type

## TL;DR

This paper improves the upper bound on the diameter of finite simple groups of Lie type, showing it grows more slowly with group size than previously known, especially for large Lie rank n.

## Contribution

It establishes a significantly tighter bound on the diameter of these groups, advancing the understanding of Babai's conjecture for unbounded Lie rank cases.

## Key findings

- Diameter bound is improved to q^{O(n (log n)^2)}
- Bound applies to subgroups of GL(V) over finite fields
- Results are tighter than previous bounds for large n

## Abstract

For a finite group $G$, let $\mathrm{diam}(G)$ denote the maximum diameter of a connected Cayley graph of $G$. A well-known conjecture of Babai states that $\mathrm{diam}(G)$ is bounded by ${(\log_{2} |G|)}^{O(1)}$ in case $G$ is a non-abelian finite simple group. Let $G$ be a finite simple group of Lie type of Lie rank $n$ over the field $F_{q}$. Babai's conjecture has been verified in case $n$ is bounded, but it is wide open in case $n$ is unbounded. Recently, Biswas and Yang proved that $\mathrm{diam}(G)$ is bounded by $q^{O( n {(\log_{2}n + \log_{2}q)}^{3})}$. We show that in fact $\mathrm{diam}(G) < q^{O(n {(\log_{2}n)}^{2})}$ holds. Note that our bound is significantly smaller than the order of $G$ for $n$ large, even if $q$ is large. As an application, we show that more generally $\mathrm{diam}(H) < q^{O( n {(\log_{2}n)}^{2})}$ holds for any subgroup $H$ of $\mathrm{GL}(V)$, where $V$ is a vector space of dimension $n$ defined over the field $F_q$.