  We show that for integers k > 1 and n > 2, the diameter of the Cayley graph
of SL_n(Z/kZ) associated to 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/kZ).
