On the McCool group $M_3$ and its associated Lie Algebra

TL;DR

This paper proves that the Lie algebra associated with the McCool group M_3 is torsion free, provides a presentation for it, and establishes that M_3 is a Magnus group, advancing understanding of its algebraic structure.

Contribution

It demonstrates the torsion freeness of the Lie algebra of M_3, offers a presentation for it, and classifies M_3 as a Magnus group, which are novel results in group theory.

Findings

Lie algebra of M_3 is torsion free

Provided a presentation for the Lie algebra of M_3

M_3 is classified as a Magnus group

Abstract

We prove that the Lie Algebra of the McCool group $M_3$ is torsion free. As a result we are able to give a presentation for the Lie Algebra of $M_3$. Furthermore, $M_3$ is a Magnus group.