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.

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.