Automorphisms of free centre-by-metabelian groups of small rank

Abstract

For a positive integer n, with $n \ge 2$, let $F_{n}$ be the free group of rank n and let $C_{n} = F_{n}/(F_{n}^{\prime \prime }, F_{n})$, that is, $C_{n}$ is a free centre-by-metabelian group of rank n. Write $\mathrm{Aut}(C_{n})$ for the automorphism group of $C_{n}$ and $T_{n}$ for the group of tame automorphisms of $C_{n}$. It has been proved by E. Stöhr (Arch Math 48:376–380, 1987) that for $2 \le n \le 3$, $\mathrm{Aut}(C_{n})$ is not finitely generated and for $n \ge 4$, $\mathrm{Aut}(C_{n})$ is generated by $T_{n}$ and one more automorphism of $C_{n}$. For $n = 2$, we find an infinite minimal subset Y of $\mathrm{Aut}(C_{2})$ such that $\mathrm{Aut}(C_{2})$ is generated by $T_{2}$ and Y. For $n = 3$, we find a subgroup of $\mathrm{Aut}(C_{3})$, generated by $T_{3}$ and two more automorphisms of $C_{3}$, which is dense in $\mathrm{Aut}(C_{3})$ with respect to the formal power series topology.