Oriented right-angled Artin pro-$\ell$ groups and maximal pro-$\ell$ Galois groups

Abstract

For a prime number $\ell$ we introduce and study oriented right-angled Artin pro-$\ell$ groups $G_{\Gamma ,\lambda }$(oriented pro-$\ell$ RAAGs for short) associated to a finite oriented graph $\Gamma$ and a continuous group homomorphism $\lambda :{\mathbb{Z}}_{\ell }\to {\mathbb{Z}}_{\ell }^{\times }$. We show that an oriented pro-$\ell$ RAAG $G_{\Gamma ,\lambda }$ is a Bloch-Kato pro-$\ell$ group if, and only if, $(G_{\Gamma ,\lambda },{\theta }_{\Gamma ,\lambda })$ is an oriented pro-$\ell$ group of elementary type generalizing a recent result of I. Snopche and P. Zalesskii. Here ${\theta }_{\Gamma ,\lambda }:G_{\Gamma ,\lambda }\to {\mathbb{Z}}_p^{\times }$ denotes the canonical $\ell$-orientation on $G_{\Gamma ,\lambda }$. We invest some effort in order to show that oriented right-angled Artin pro-$\ell$ groups share many properties with right-angled Artin pro-$\ell$-groups or even discrete RAAG's, e.g., if $\Gamma$ is a specially oriented chordal graph, then $G_{\Gamma ,\lambda }$ is coherent, generalizing a result of C. Droms. Moreover, in this case $(G_{\Gamma ,\lambda },{\theta }_{\Gamma ,\lambda })$ has the Positselski-Bogomolov property generalizing a result of H. Servatius, C. Droms and B. Servatius for discrete RAAG's. If $\Gamma$ is a specially oriented chordal graph and $\mathrm{Im}(\lambda )\subseteq 1+4{\mathbb{Z}}_2$ in case that $\ell =2$, then ${\mathrm{H}}^{\bullet }(G_{\Gamma ,\lambda },{\mathbb{F}}_{\ell })\simeq {\Lambda }^{\bullet }({\ddot{\Gamma }}^{\mathrm{op}})$ generalizing a well known result of M. Salvetti.

The differences between the 1st version (Apr'23) and the 2nd are: correction of a couple of minor misprints, dedication to the memory of Avinoam Mann