Diameter of classical groups generated by transvections

Let $G$ be a finite classical group generated by transvections, i.e., one of ${\mathrm{SL}}_n(q)$, ${\mathrm{SU}}_n(q)$, ${\mathrm{Sp}}_{2n}(q)$, or ${\mathrm{O}}_{2n}^{\pm }(q)$ ($q$ 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 $\mathrm{Cay}(G,X)$ 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 ${\mathrm{SL}}_n(q)$, ${\mathrm{SU}}_n(q)$, ${\mathrm{Sp}}_{2n}(q)$ and $X$ contains three random generators then with high probability the diameter $\mathrm{Cay}(G,X)$ 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.