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 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 Fn/R′ where Fn is a free group of finite rank n at least two and R a characteristic subgroup of Fn.