Diameters of Cayley graphs of Chevalley groups

We show that for integers k ≥ 2 and n ≥ 3 , the diameter of the Cayley graph of SL n ( Z / k Z ) associated with a standard two-element generating set is at most a constant times n 2 ln k . This answers a question of A. Lubotzky concerning SL n ( F p ) and is unexpected because these Cayley graphs do not form an expander family. Our proof amounts to a quick algorithm for finding short words representing elements of SL n ( Z / k Z ) . We generalize our results to other Chevalley groups over Z / k Z .