Published online by Cambridge University Press: 08 December 2014
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 $\mathfrak{p}$Mn of the derivation algebra of the Chen Lie algebra. Furthermore, we completely determine the Johnson images, and give the abelianisation of
$\mathfrak{p}$Mn as a Lie algebra by using Morita's trace maps.