Diameter of classical groups generated by transvections

TL;DR

This paper proves that the diameter of Cayley graphs of classical groups generated by transvections is polynomially bounded, confirming Babai's conjecture for these groups and certain generating sets, with implications for random generators.

Contribution

It establishes a polynomial bound on the diameter of Cayley graphs of classical groups generated by transvections, confirming Babai's conjecture in this context.

Findings

Diameter bounded by (n log q)^C for some constant C

High probability diameter bound of n^{O(log q)} with three random generators

Confirms Babai's conjecture for classical groups over small fields

Abstract

Let $G$ be a finite classical group generated by transvections, i.e., one of $\operatorname{SL}_n(q)$, $\operatorname{SU}_n(q)$, $\operatorname{Sp}_{2n}(q)$, or $\operatorname{O}^\pm_{2n}(q)$ ($q$ even), and let $X$ be a generating set for $G$ containing at least one transvection. Building on work of Garonzi, Halasi, and Somlai, we prove that the diameter of the Cayley graph $\operatorname{Cay}(G, X)$ is bounded by $(n \log q)^C$ for some constant $C$. This confirms Babai's conjecture on the diameter of finite simple groups in the case of generating sets containing a transvection. By combining this with a result of the author and Jezernik it follows that if $G$ is one of $\operatorname{SL}_n(q)$, $\operatorname{SU}_n(q)$, $\operatorname{Sp}_{2n}(q)$ and $X$ contains three random generators then with high probability the diameter $\operatorname{Cay}(G, X)$ is bounded by $n^{O(\log q)}$. This confirms Babai's conjecture for non-orthogonal classical simple groups over small fields and three random generators.