Babai's conjecture for high-rank classical groups with random generators

TL;DR

This paper proves that for large classical groups generated by many random elements, the Cayley graph's diameter is polynomially bounded with high probability, advancing understanding of group expansion and Babai's conjecture.

Contribution

It establishes diameter bounds for Cayley graphs of high-rank classical groups with random generators, including the case of a fixed number of generators for special groups.

Findings

Diameter bounded by $q^2 n^{O(1)}$ with high probability

For $G=SL_n(p)$ with fixed prime $p$, diameter bound holds with 3 generators

Results support Babai's conjecture for high-rank classical groups

Abstract

Let $G = \mathrm{SCl}_n(q)$ be a quasisimple classical group with $n$ large, and let $x_1, \dots, x_k \in G$ random, where $k \geq q^C$. We show that the diameter of the resulting Cayley graph is bounded by $q^2 n^{O(1)}$ with probability $1 - o(1)$. In the particular case $G = \mathrm{SL}_n(p)$ with $p$ a prime of bounded size, we show that the same holds for $k = 3$.