Small conjugacy classes in the automorphism groups of relatively free groups

Abstract

a b s t r a c tLet F be an infinitely generated free group and let R be a fully invariant subgroup of F such that (a) R is contained in the commutator subgroup F ′ of F and (b) the quotient group F /R is residually torsion-free nilpotent. Then the automorphism group Aut(F /R ′ ) of the group F /R ′ is complete. In particular, the automorphism group of any infinitely generated free solvable group of derived length at least two is complete.This extends a result by Dyer and Formanek (1977) [7] on finitely generated groups F n /R ′ where F n is a free group of finite rank n at least two and R a characteristic subgroup of F n .

The Small Index Property for Free Nilpotent Groups

“…Proof. The result is proved in [14] for infinitely generated relatively free groups, and in fact the proof given in [14] can be used without significant changes in the general case. For the reader's convenience, we reproduce the plan of the proof.…”

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

“…Theorem 9 (Tolstykh [21]). Let F ∞ be an infinitely generated free group, R ≤ F ′ ∞ a fully characteristic subgroup of F ∞ such that the quotient group F ∞ /R is residually torsion-free nilpotent.…”

Local–global invariants of finite and infinite groups: Around Burnside from another side

“…As to infinite groups, typical examples of complete groups arise as groups of automorphisms G = Aut (F ), where F is a free group (or a group that is "not so far" from free). The cases where F is a free group, free nilpotent group of class two, or the quotient of a free group by an appropriate characteristic subgroup were treated, respectively, in [DF1]- [DF3] (for groups of finite rank) and [To1]- [To3] (for groups of infinite rank). The cases F = GL(n, Z) (n odd), F = PGL(2, Z) (n ≥ 2), F = SL(n, Z) (n ≥ 3) were considered in [Dy].…”