Quotient groups of IA-automorphisms of free metabelian groups

Abstract

For a positive integer $n$, with $n \geq 2$, 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 $c \geq 2$, let ${\rm I}_{c}{\rm A}(M_n)$ be the subgroup of ${\rm Aut}(M_{n})$ consisting of all automorphisms inducing the identity mapping on $M_n/\gamma_c(M_n)$. In this paper, we study the quotient groups ${\cal L}^{c}({\rm IA}(M_{n})) = {\rm I}_{c}{\rm A}(M_n)/{\rm I}_{c+1}{\rm A}(M_n)$ for all $n$ and $c$. For $c \geq 2$, we show $\gamma_{c}({\rm IA}(M_{2})) = {\rm I}_{c+1}{\rm A}(M_{2}))$. For $n = 3$, we show $\gamma_{3}({\rm IA}(M_{3})) \neq {\rm I}_{4}{\rm A}(M_{3})$ and so, the Andreadakis' conjecture (for a free metabelian group) is not valid for $n = 3$ and $c = 3$. For $n \geq 4$ and $c \geq 3$, we prove that ${\cal L}^{c}({\rm IA}(M_{n})) = \gamma_{c-1}({\rm IA}(M_{n})){\rm I}_{c+1}{\rm A}(M_{n})/{\rm I}_{c+1}{\rm A}(M_{n})$.