Small conjugacy classes in the automorphism groups of relatively free groups

Let 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., is a free group of finite rank flat least two and R a characteristic subgroup of F(n). (C) 2010 Elsevier B.V. All rights reserved.