Quotient groups of IA-automorphisms of free metabelian groups

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 ${\rm I}{c}{\rm A}(M_n)$bethesubgroupof${\rm Aut}(M{n})$ consisting of all automorphisms inducing the identity mapping on $M_n\mathrm{/}{\gamma }_c(M_n)$. In this paper, we study the quotient groups ${\cal L}^{c}({\rm IA}(M_{n})) = {\rm I}{c}{\rm A}(M_n)/{\rm I}{c+1}{\rm A}(M_n)$forall$n$and$c$\mathrm{.}For$c \geq 2$,weshow$\gamma_{c}({\rm IA}(M_{2})) = {\rm I}{c+1}{\rm A}(M{2}))$\mathrm{.}For$n = 3$, we show ${\gamma }_3(\mathrm{I}\mathrm{A}(M_3))\mathrm{\ne }{\mathrm{I}}_4\mathrm{A}(M_3)$ 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 ${\cal L}^{c}({\rm IA}(M_{n})) = \gamma_{c-1}({\rm IA}(M_{n})){\rm I}{c+1}{\rm A}(M{n})/{\rm I}{c+1}{\rm A}(M{n})$.