Quotient groups of IA-automorphisms of free metabelian groups

TL;DR

This paper investigates the structure of quotient groups derived from IA-automorphisms of free metabelian groups, revealing cases where the Andreadakis' conjecture holds or fails, and providing explicit descriptions of these quotients.

Contribution

It establishes new results on the quotient groups of IA-automorphisms of free metabelian groups, including counterexamples to the Andreadakis' conjecture for certain ranks.

Findings

For n=2, the lower central series of IA(M_2) matches the filtration I_cA(M_2).

For n=3, the Andreadakis' conjecture does not hold at c=3.

For n≥4 and c≥3, explicit descriptions of the quotient groups are provided.

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 ${\cal L}^{c}({\rm IA}(M_{n})) = \gamma_{c-1}({\rm IA}(M_{n})){\rm I}_{c+1}{\rm A}(M_{n})/{\rm I}_{c+1}{\rm A}(M_{n})$.