# Quotient groups of IA-automorphisms of a free group of rank 3

## TL;DR

This paper investigates the structure of quotient groups derived from the McCool group of rank 3, revealing their relation to free groups and conditions for embedding into Johnson Lie algebras.

## Contribution

It establishes an isomorphism between certain quotients of the McCool group and free groups, and characterizes when the associated graded Lie algebra embeds into the Johnson Lie algebra.

## Key findings

- Quotients of the McCool group are isomorphic to two copies of free group quotients.
- Provides a necessary and sufficient condition for embedding the graded Lie algebra into the Johnson Lie algebra.
- Enhances understanding of the algebraic structure of IA-automorphisms of free groups.

## Abstract

We prove that, for any positive integer $c$, the quotient group $\gamma_{c}(M_{3})/\gamma_{c+1}(M_{3})$ of the lower central series of the McCool group $M_{3}$ is isomorphic to two copies of the quotient group $\gamma_{c}(F_{3})/\gamma_{c+1}(F_{3})$ of the lower central series of a free group $F_{3}$ of rank $3$ as $\mathbb{Z}$-modules. Furthermore, we give a necessary and sufficient condition whether the associated graded Lie algebra ${\rm gr}(M_{3})$ of $M_3$ is naturally embedded into the Johnson Lie algebra ${\cal L}({\rm IA}(F_{3}))$ of the IA-automorphisms of $F_{3}$.