Cohomology of the pure symmetric automorphisms of right-angled Artin groups

Abstract: We compute the cohomology groups of the pure symmetric outer automorphism group $Σ$POut$(A_Γ)$ and the pure symmetric automorphism group $Σ$PAut$(A_Γ)$ of a right-angled Artin group $A_Γ$. Using the equivariant spectral sequence arising from the action of $Σ$POut$(A_Γ)$ on the generalized McCullough-Miller complex MM$Γ$, we show that $H^q(Σ$POut$(AΓ))$ is free abelian and we compute its rank in terms of the combinatorics of certain poset. Applying the Lyndon-Hochschild-Serre spectral sequence and the Leray-Hirsch theorem we do the same for $H^q(Σ$PAut$(A_Γ))$. In both cases the cohomology ring is generated in degree 1. Finally, we introduce the Generalized Brownstein-Lee Conjecture, proposing a presentation of $H^*(Σ$PAut$(A_Γ))$, and prove that it holds in dimension $2$.