Induced automorphisms of free metabelian groups

Abstract

Let $F$ and $\phi$ respectively be the free group and the free metabelian group of rank $q$. Let ${\phi _n}$ be the $n$th group of the lower central series of $\phi$. We show that, for $q > 3$ and for any positive integer $n$, every automorphism of $\phi /{\phi _n}$, which is induced by an automorphism of $\phi$, is induced by an automorphism of $F$. This reopens the question of whether every automorphism of $\phi$ is induced by an automorphism of $F$ if $q > 3$.