The Automorphism Tower of a Free Group

We prove that the automorphism group of any non-abelian free group F is complete. The key technical step in the proof: the set of all conjugations by powers of primitive elements is first-order parameter-free definable in the group Aut(F ). Introduction In 1975 J. Dyer and E. Formanek [2] had proved that the automorphism group of a finitely generated non-abelian free group F is complete (that is, it is centreless and all its automorphisms are inner) and so Aut(Aut(F )) ∼= Aut(F ). They noted that their research was stimulated by G. Baumslag, who conjectured that the automorphism tower of a finitely generated free group is very short. New proofs for the result of Dyer and Formanek were given in 1990 by D. G. Khramtsov [6] and E. Formanek [4]. The objective of this paper is to generalize the result of Dyer and Formanek from finitely generated non-abelian free groups to arbitrary non-abelian free groups. Let F be a free non-abelian group. We obtain a group-theoretic characterization of conjugations by powers of primitive elements in Aut(F ). Our key technical results can be summarized in model-theoretic terms as follows: the set of all conjugations by powers of primitive elements is first-order parameter-free definable in the group Aut(F ) (Theorem 5.1). The latter means that there is a first-order formula with one free variable in the language of groups such that its realizations in Aut(F ) are exactly conjugations just mentioned. Therefore the subgroup of all conjugations (inner automorphisms of F ) is a characteristic subgroup of Aut(F ). This implies that the group Aut(F ) is complete (Theorem 5.4). The main technical tool in the proof of Theorem 5.1 is the use of conjugacy classes of involutions based on a characterization of involutions in Aut(F ) given by J. Dyer and G. P. Scott in [3]. An important role is played by involutions of the following sort. Let x be a primitive element of F and F = 〈x〉 ∗C a free factorization of F. Then an automorphism of F which inverts x and takes each element in C to its conjugate by x, is an involution. We call any involution obtained in such a way a quasi-conjugation, since it acts as conjugation on a ‘large’ subgroup of F. Date: September 19, 1997. 1991 Mathematics Subject Classification. 20F28 (20E05, 03C60). Supported by Russian Foundation of Fundamental Research Grant 96-01-00456. 1