Right-angled Artin groups as finite-index subgroups of their outer automorphism groups

TL;DR

This paper demonstrates that every right-angled Artin group can be embedded as a finite-index subgroup within the outer automorphism group of another right-angled Artin group, with explicit constructions and conditions for such embeddings.

Contribution

It establishes that all right-angled Artin groups appear as finite-index subgroups of automorphism groups of other right-angled Artin groups, providing explicit constructions and conditions.

Findings

Every right-angled Artin group is a finite-index subgroup of an automorphism group.

Explicit constructions using pure symmetric outer automorphisms are provided.

Conditions by Day-Wade and Wade-Brück determine when these automorphism groups are right-angled Artin groups and have finite index.

Abstract

We prove that every right-angled Artin group occurs as a finite-index subgroup of the outer automorphism group of another right-angled Artin group. We furthermore show that the latter group can be chosen in such a way that the quotient is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^N$ for some $N$. For these, we give explicit constructions using the group of pure symmetric outer automorphisms. Moreover, we need two conditions by Day-Wade and Wade-Br\"uck about when this group is a right-angled Artin group and when it has finite index.