Infinitely generated free nilpotent groups: completeness of the automorphism groups

TL;DR

This paper proves that the automorphism groups of infinitely generated free nilpotent groups of class at least two are complete, confirming that their automorphism towers terminate after finitely many steps, thus resolving Baumslag's conjecture.

Contribution

It establishes the completeness of automorphism groups for all infinitely generated free nilpotent groups of class at least two, extending previous results and confirming the finite termination of their automorphism towers.

Findings

Automorphism groups of infinitely generated free nilpotent groups are complete.

Automorphism towers of these groups terminate after finitely many steps.

Confirms Baumslag's conjecture for infinitely generated cases.

Abstract

Baumslag conjectured in the 1970s that the automorphism tower of a finitely generated free nilpotent group must be very short. Let F_{n,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 F_{n,2} is even complete (and hence the height of the aumorphism tower of F_{n,2} is two) provided that n is not three; in the case when n=3, the height of the automorphism tower of F_{n,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 F_{n,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 F_{n,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.