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 $\lambda\colon\mathbb Z_\ell\to\mathbb Z_\ell^\times$. 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 $\theta_{\Gamma,\lambda}\colon G_{\Gamma,\lambda}\to\mathbb Z_p^\times$ 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 ${\rm Im}(\lambda)\subseteq 1+4\mathbb Z_2$ in case that $\ell=2$, then ${\rm H}^\bullet(G_{\Gamma,\lambda},\mathbb F_\ell) \simeq \Lambda^\bullet(\ddot{\Gamma}^{\rm op})$ generalizing a well known result of M. Salvetti.