# IA-automorphisms and Lie Algebras related to the McCool group

## TL;DR

This paper studies a subgroup of the McCool group, proving it has a solvable conjugacy problem, analyzing its associated Lie algebra, and establishing its embedding into the IA automorphisms' Lie algebra.

## Contribution

It introduces and analyzes the subgroup $I_n$ of the McCool group, providing its Lie algebra presentation and embedding into the Andreadakis-Johnson Lie algebra.

## Key findings

- $I_n$ has a solvable conjugacy problem
- Presented a Lie algebra for $I_n$
- Embedded gr($I_n$) into the IA automorphisms' Lie algebra

## Abstract

In the present work we investigate a subgroup $I_n$ of the McCool group $M_n$. We show that $I_n$ has solvable conjugacy problem. Next, we investigate its Lie Algebra gr($I_n$) and we find a presentation for it. Finally we show that gr($I_n$) is naturally embedded into the Andreadakis-Johnson Lie Algebra of the IA automorphisms of the free group $F_n$.