On automorphisms of certain free nilpotent-by-abelian Lie algebras

Abstract

For a positive integer $n\ge 4$, let $R_n$ be a free (nilpotent of class 2)-by-abelian and abelian-by-(nilpotent of class 2) Lie algebra of rank n. We show that the subgroup of $\mathrm{\text{Aut}}(R_n)$ generated by the tame automorphisms and a countably infinite set of explicitly given automorphisms of $R_n$ is dense in $\mathrm{\text{Aut}}(R_n)$ with respect to the formal power series topology.