Representations of pure symmetric automorphism groups of RAAGs

Abstract

We study representations of the pure symmetric automorphism group PAut(AГ) of a RAAG AГ with defining graph Г. We first construct a homomorphism from PAut(AГ) to the direct product of a RAAG and a finite direct product of copies of F2 × F2; moreover, the image of PAut(AГ) under this homomorphism is surjective onto each factor. As a consequence, we obtain interesting actions of PAut(AГ) on non-positively curved spaces We then exhibit, for connected Г, a RAAG which properly contains Inn(AГ) and embeds as a normal subgroup of PAut(AГ). We end with a discussion of the linearity problem for PAut(AГ).