Growth estimates and diameter bounds for untwisted classical groups

Babai's conjecture states that, for any finite simple non-abelian group $G$, the diameter of $G$ is bounded by $(\log|G|)^{C}$ for some absolute constant $C$. We prove that, for any untwisted classical group $G$ of rank $r$ defined over a field $\mathbb{F}_{q}$ with $q$ not too small with respect to $r$, \begin{equation*} \mathrm{diam}(G(\mathbb{F}_{q}))\leq(\log|G(\mathbb{F}_{q})|)^{408r^{4}}. \end{equation*} This bound improves on results by Breuillard, Green, and Tao [9], Pyber and Szab\'o [38], and, for $q$ large enough, also by Halasi, Mar\'oti, Pyber, and Qiao [16]. Our approach is in several ways closer to that of preexistent work by Helfgott [20], in that we give dimensional estimates (that is, bounds of the form $|A\cap V(\mathbb{F}_{q})|\ll|A^{C}|^{\dim(V)/\dim(G)}$, where $A$ is any generating set) for varieties $V$ of specific types, and work in the Lie algebra whenever possible. One of our main tools is a new, more efficient form of escape from subvarieties.