Diameter Bound for Finite Simple Groups of Large Rank

TL;DR

This paper establishes an upper bound on the diameter of Cayley graphs for large-rank finite simple groups of Lie type, relating it to the group's rank and field size, advancing understanding of their expansion properties.

Contribution

It provides a new diameter bound for finite simple groups of Lie type with large rank, extending previous conjectures and results in group theory.

Findings

Diameter bound of $q^{O(n( ext{log } n + ext{log } q)^3)}$ for groups of rank n

Progress towards Babai's conjecture for large-rank groups

Enhanced understanding of Cayley graph expansion in Lie type groups

Abstract

Given a non-abelian finite simple group $G$ of Lie type, and an arbitrary generating set $S$, it is conjectured by Laszlo Babai that its Cayley graph $\Gamma (G,S)$ will have a diameter of $(\log |G|)^{O(1)}$. However, little progress has been made when the rank of $G$ is large. In this article, we shall show that if $G$ has rank $n$, and its base field has order $q$, then the diameter of $\Gamma (G,S)$ would be $q^{O(n(\log n + \log q)^3)}$.