Infinitely generated free nilpotent groups: completeness of the automorphism groups
Abstract
Baumslag conjectured in the 1970s that the automorphism tower of a finitely generated free nilpotent group must be very short. Let Fn,c denote a free nilpotent group of finite rank n at least two and of nilpotency class c at least two. In 1976 Dyer and Formanek proved that the automorphism group of Fn,2 is even complete (and hence the height of the aumorphism tower of Fn,2 is two) provided that n is not three; in the case when n=3, the height of the automorphism tower of Fn,2 is three. The author proved in 2001 that the automorphism group of any infinitely generated free nilpotent of class two is complete. In his Ph. D. thesis (2003) Kassabov found an upper bound u(n,c) (a natural number) for the height of the automorphism tower of Fn,c in terms of n and c, thereby finally proving Baumslag's conjecture. By analyzing the function u(n,c), one can conclude that if c is small compared to n, then the height of the automorphism tower of Fn,c is at most three. The main result of the present paper states that the automorphism group of any infinitely generated free nilpotent group of nilpotency class at least two is complete. Thus the automorphism tower of any free nilpotent group terminates after finitely many steps.