Pure symmetric automorphisms, extensions of RAAGs, and Koszulness

Abstract

We characterize in terms of a combinatorial condition on the graph $\Gamma$ when the group $\mathrm{PAut}(A_\Gamma)$ of pure symmetric automorphisms of the RAAG $A_\Gamma$ and its outer version $\mathrm{POut}(A_\Gamma)$ have a descending central Lie algebra which is Koszul. To do that, we prove that our combinatorial condition implies that these groups are iterated extensions of RAAGs; in particular, they are poly-free. On the other hand, we show that $\mathrm{PAut}(F_n)$ is not poly-finitely generated free for $n \geq 4$. We also show that groups in a certain class containing $\mathrm{PAut}(A_\Gamma)$ are 1-formal.