ON THE BERGMAN PROPERTY FOR THE AUTOMORPHISM GROUPS OF RELATIVELY FREE GROUPS

A group G is said to have the Bergman property (the property of uniformity of finite width) if given any generating X with X = X−1 of G, we have that G = Xk for some natural k, that is, every element of G is a product of at most k elements of X. We prove that the automorphism group Aut(N) of any infinitely generated free nilpotent group N has the Bergman property. Also, we obtain a partial answer to a question posed by Bergman by establishing that the automorphism group of a free group of countably infinite rank is a group of uniformly finite width.