Girth, words and diameter

We study the girth of Cayley graphs of finite classical groups (Formula presented.) on random sets of generators. Our main tool is an essentially best possible bound we obtain on the probability that a given word (Formula presented.) takes the value 1 when evaluated in (Formula presented.) in terms of the length of (Formula presented.), which has additional applications. We also study the girth of random directed Cayley graphs of symmetric groups, and the relation between the girth and the diameter of random Cayley graphs of finite simple groups.