Small conjugacy classes in the automorphism groups of relatively free groups

TL;DR

This paper proves that automorphism groups of certain infinitely generated free groups and their quotients are complete, extending previous results from finitely generated cases to more general infinite settings.

Contribution

It generalizes the completeness of automorphism groups from finitely generated free groups to infinitely generated cases under specific conditions.

Findings

Automorphism groups of certain infinitely generated free groups are complete.

Extension of Dyer and Formanek's finite case result to infinite cases.

Automorphism groups of free solvable groups of derived length ≥ 2 are complete.

Abstract

Let F be an infinitely generated free group and R 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 solvalbe group of derived length at least two is complete. This extends a result by Dyer and Formanek (1977) 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.