Quotient groups of IA-automorphisms of free metabelian groups

Abstract

For a positive integer $n$, with $n\ge 2$, let $M_n$ be a free metabelian group of rank $n$. For $c\in \mathbb{N}$, let ${\gamma }_c(M_n)$ be the $c$-th term of the lower central series of $M_n$. For $c\ge 2$, let ${\mathrm{I}}_c\mathrm{A}(M_n)$ be the subgroup of $\mathrm{Aut}(M_n)$ consisting of all automorphisms inducing the identity mapping on $M_n/{\gamma }_c(M_n)$. In this paper, we study the quotient groups ${\mathcal{L}}^c(\mathrm{IA}(M_n))={\mathrm{I}}_c\mathrm{A}(M_n)/{\mathrm{I}}_{c+1}\mathrm{A}(M_n)$ for all $n$ and $c$. For $c\ge 2$, we show ${\gamma }_c(\mathrm{IA}(M_2))={\mathrm{I}}_{c+1}\mathrm{A}(M_2))$. For $n=3$, we show ${\gamma }_3(\mathrm{IA}(M_3))\ne {\mathrm{I}}_4\mathrm{A}(M_3)$ and so, the Andreadakis' conjecture (for a free metabelian group) is not valid for $n=3$ and $c=3$. For $n\ge 4$ and $c\ge 3$, we prove that ${\mathcal{L}}^c(\mathrm{IA}(M_n))={\gamma }_{c-1}(\mathrm{IA}(M_n)){\mathrm{I}}_{c+1}\mathrm{A}(M_n)/{\mathrm{I}}_{c+1}\mathrm{A}(M_n)$.