Infinitely generated free nilpotent groups: completeness of the automorphism groups

Abstract

Abstract. We transfer the results of Dyer, Formanek and Kassabov on the automorphism towers of finitely generated free nilpotent groups to infinitely generated free nilpotent groups. We prove that the automorphism groups of infinitely generated free nilpotent groups are complete. By combining the results of Dyer, Formanek, Kassabov with the results in the present paper, one gets that the automorphism tower of any free nilpotent group terminates after finitely many steps. IntroductionBaumslag conjectured in the 1970s that the automorphism tower of a finitely generated free group (free nilpotent group) must be very short. Dyer and Formanek [8] justified the conjecture concerning finitely generated free groups in the "sharpest sense" by proving that the automorphism group Aut(F n ) of a non-abelian free group F n of finite rank n is complete. Recall that a group G is said to be complete if G is centreless and all automorphisms of G are inner; it then follows that Aut(G) ∼ = G. Thus Aut(Aut(F n )) ∼ = Aut(F n ), or, in other words, the height of the automorphism tower over F n is two. The proof of completeness of Aut(F n ) given by Dyer and Formanek in [8] has later been followed by the proofs given by Formanek [11], by Khramtsov [14], by Bridson and Vogtmann [3], and by the author [17]. The proof given in [17] works for arbitrary non-abelian free groups; thus the automorphism groups of infinitely generated free groups are also complete.Let F n,c denote a free nilpotent group of finite rank n 2 and of nilpotency class c 2. In [9] Dyer and Formanek studied the automorphism towers of free nilpotent groups F n,2 of class two. They showed that the group Aut(F n,2 ) is complete provided that n = 3. In the case when n = 3 the height of the automorphism tower of F n,2 is three. The main result of [18] states that the automorphism group of any infinitely generated free nilpotent group of class two is complete.In [10] Dyer and Formanek proved completeness of the automorphism groups of groups of the form F n /R ′ where R is a characteristic subgroup of F n which is contained in the commutator subgroup F ′ n of F n and F n /R is residually torsion-free nilpotent.In his Ph. D. thesis [13] Kassabov found an upper bound u(n, c) ∈ N for the height of the automorphism tower of F n,c in terms of n and c, thereby finally proving Baumslag's conjecture on finitely generated free nilpotent groups. 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.2000 Mathematics Subject Classification. 20F28 (20F18, 03C60).

On the automorphism groups of relatively free groups of infinite rank: a survey

“…• The subgroup Inn(F ∞ ) is then first-order definable in Aut(F ∞ ). Theorem 8 (Tolstykh [17], [18]). For any k ≥ 2, the automorphism group Aut(N ∞,k ) of any free nilpotent group N ∞,k of infinite rank is complete.…”