Short laws for finite groups of Lie type

We produce new short laws in two variables valid in finite groups of Lie type. Our result improves upon results of Kozma and the second named author, and is sharp up to logarithmic factors, for all families except possibly the Suzuki groups. We also produce short laws valid for generating pairs and random pairs in finite groups of Lie type, and, conditional on Babai’s diameter conjecture, make effective the dependence of our bounds on the rank. Our proof uses, among other tools, the classification of finite simple groups, Aschbacher’s structure theorem for maximal subgroups for classical groups, and upper bounds on the diameters of finite simple groups due to Breuillard, Green, Guralnick, Pyber, Szabó and Tao.