Diameter of classical groups generated by transvections

Let G be a finite classical group generated by transvections, i.e., one of SLn(q), SUn(q), Sp2n(q), or O2n±(q)(qeven), 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 Cay(G,X) is bounded by (nlog⁡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 SLn(q), SUn(q), Sp2n(q) and X contains three random generators then with high probability the diameter Cay(G,X) is bounded by nO(log⁡q). This confirms Babai's conjecture for non-orthogonal classical simple groups over small fields and three random generators.