Representations of pure symmetric automorphism groups of RAAGs

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 $PAut(A_\Gamma)$.