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

TL;DR

This paper computes the cohomology groups of pure symmetric automorphism groups of right-angled Artin groups, revealing their structure and proposing a new conjecture about their presentations.

Contribution

It introduces methods to compute cohomology of these automorphism groups and proves the Generalized Brownstein-Lee Conjecture in dimension 2.

Findings

Cohomology groups are free abelian with ranks determined by combinatorics.

Cohomology ring is generated in degree 1.

The Generalized Brownstein-Lee Conjecture holds in dimension 2.

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 $H^*(\Sigma$PAut$(A_\Gamma))$, and prove that it holds in dimension $2$.