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 = X k 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.