# On embeddings between outer automorphism groups of right-angled Artin groups

## TL;DR

This paper constructs new embeddings between outer automorphism groups of right-angled Artin groups using abelian coverings, explores their properties, and investigates finite subgroups and non-embeddability conditions.

## Contribution

It introduces novel embeddings of outer automorphism groups via abelian coverings and analyzes their subgroup structures and embeddability constraints.

## Key findings

- Constructed embeddings of outer automorphism groups using abelian coverings.
- Determined $p$-adic valuations and $bZ_p$-ranks of automorphism groups.
- Identified non-embeddability conditions based on group invariants.

## Abstract

Using abelian coverings of Salvetti complexes, embeddings of outer automorphism groups of right-angled Artin groups (RAAGs) into outer automorphism groups of their particular characteristic subgroups are constructed. Virtual embeddings of outer automorphism groups of finitely generated groups having the unique root property into outer automorphism groups of their particular subgroups are also given. These results provide us with rich examples of (virtual) embeddings between outer automorphism groups of RAAGs. Finite subgroups of pure (outer) automorphism groups of RAAGs are also investigated. $p$-adic valuations and $\mathbb{Z}_p$-ranks of these groups are determined to establish some non-embeddability conditions.