Representations of pure symmetric automorphism groups of RAAGs

TL;DR

This paper explores the structure and representations of the pure symmetric automorphism groups of right-angled Artin groups, constructing homomorphisms to known groups, analyzing actions on geometric spaces, and discussing linearity properties.

Contribution

It introduces a homomorphism from the automorphism group to a product of RAAGs and free groups, and investigates subgroup embeddings and linearity questions.

Findings

Constructed a surjective homomorphism to a product of RAAGs and free groups.

Established actions of the automorphism group on non-positively curved spaces.

Identified a RAAG embedding as a normal subgroup of the automorphism group.

Abstract

We study representations of the pure symmetric automorphism group $PAut(A_\Gamma)$ of a RAAG $A_\Gamma$ with defining graph $\Gamma$. We first construct a homomorphism from $PAut(A_\Gamma)$ to the direct product of a RAAG and a finite direct product of copies of $F_2 \times F_2$; moreover, the image of $PAut(A_\Gamma)$ under this homomorphism is surjective onto each factor. As a consequence, we obtain interesting actions of $PAut(A_\Gamma)$ on non-positively curved spaces We then exhibit, for connected $\Gamma$, a RAAG which property contains $Inn(A_\Gamma)$ and embeds as a normal subgroup of $PAut(A_\Gamma)$. We end with a discussion of the linearity problem for $PAut(A_\Gamma)$.