# A note on nilpotent subgroups of automorphism groups of RAAGs

## TL;DR

This paper shows that automorphism groups of right-angled Artin groups can contain non-abelian nilpotent subgroups, specifically the Heisenberg group, under certain conditions involving adjacent transvections.

## Contribution

It extends previous results by demonstrating the presence of the Heisenberg group in automorphism groups of RAAGs when specific elements are present.

## Key findings

- Automorphism groups of RAAGs can contain the Heisenberg group.
- Presence of adjacent transvections is key to this property.
- Extension of Charney-Vogtmann's earlier results.

## Abstract

We observe that automorphism groups of right-angled Artin groups contain nilpotent non-abelian subgroups, namely $H_3(\mathbb{Z})$ the three-dimensional integer Heisenberg group, provided they admit a certain type of element, called an adjacent transvection. This represents a (minor) extension of a result of Charney-Vogtmann.