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

Abstract

We compute the cohomology groups of the pure symmetric outer automorphism group $\Sigma$POut$(A_{\Gamma })$ and the pure symmetric automorphism group $\Sigma$PAut$(A_{\Gamma })$ of a right-angled Artin group $A_{\Gamma }$. Using the equivariant spectral sequence arising from the action of $\Sigma$POut$(A_{\Gamma })$ on the generalized McCullough-Miller complex MM${}_{\Gamma }$, we show that $H^q(\Sigma$POut$(A_{\Gamma }))$ 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 $H^q(\Sigma$PAut$(A_{\Gamma }))$. In both cases the cohomology ring is generated in degree 1. Finally, we introduce the Generalized Brownstein-Lee Conjecture, proposing a presentation of $H^{\ast }(\Sigma$PAut$(A_{\Gamma }))$, and prove that it holds in dimension $2$.