On the basis-conjugating automorphism groups of free groups and free metabelian groups

In this paper we study the images of the Johnson homomorphisms of the basis-conjugating automorphism groups of free groups and free metabelian groups. In particular, we show that the Johnson image is contained in a certain proper Lie subalgebra p(n)(M) of the derivation algebra of the Chen Lie algebra. Furthermore, we completely determine the Johnson images, and give the abelianisation of p(n)(M) as a Lie algebra by using Morita's trace maps.