Infinitely Generated Free Nilpotent Groups: Completeness of the Automorphism Groups

Baumslag conjectured in the 1970s that the automorphism tower of a finitely generated free group (free nilpotent group) must be very short. Dyer and Formanek [9] justified the conjecture concerning finitely generated free groups in the “sharpest sense” by proving that the automorphism group Aut(Fn) of a non-abelian free group Fn 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(Fn)) ∼= Aut(Fn), or, in other words, the height of the automorphism tower over Fn is two. The proof of completeness of Aut(Fn) given by Dyer and Formanek in [9] have later been followed by the proofs given by Formanek [12], by Khramtsov [15], by Bridson and Vogtmann [3], and by the author [18]. The proof given in [18] works for arbitrary non-abelian free groups; thus the automorphism groups of infinitely generated free groups are also complete. Let Fn,c denote a free nilpotent group of finite rank n > 2 and of nilpotency class c > 2. In [10] Dyer and Formanek studied the automorphism towers of free nilpotent groups Fn,2 of class two. They showed that the group Aut(Fn,2) is complete provided that n 6= 3. In the case when n = 3 the height of the automorphism tower of Fn,2 is three. The main result of [19] states that the automorphism group any infinitely generated free nilpotent group of class two is complete. In [11] Dyer and Formanek proved completeness of the automorphism groups of groups of the form Fn/R ′ where R is a characteristic subgroup of Fn which is contained in the commutator subgroup F ′ n of Fn, and Fn/R is residually torsionfree nilpotent. It follows, in particular, that the automorphism group of any finitely generated non-abelian free solvable group is complete.