On Bergman's property for the automorphism groups of relatively free groups

Abstract

We say that a group $G$ has Bergman's property (the property of universality of finite width) if for every generating set $X$ of $G$ with $X=X^{-1}$ we have that $G=X^k$ for some natural number $k.$ The property is named after George Bergman who have proved recently that the infinite symmetric groups are groups of universally finite width. The first example of an infinite group with Bergman's property is due to Shelah (1980s). Lately some other examples have been found: the automorphism groups of doubly transitive chains (Droste-Holland), the automorphism group of reals as a Borel space (Droste-G\"obel) and the infinite-dimensional general linear groups over division rings. In the present paper we prove that the automorphism group $Aut(N)$ of any infinitely generated free nilpotent group $N$ has Bergman's property, is generated by involutions, perfect and has confinality greater than $rank(N).$ Also, we obtain a partial answer to a question posed by Bergman establishing that the automorphism group of a free group of countably infinite rank is a group of universally finite width.