Oriented Right-Angled Artin Pro-ℓ Groups and Maximal Pro-ℓ Galois Groups

Abstract

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. Snopce and P. Zalesskiĭ—here $\theta _{\Gamma ,\lambda }\colon G_{\Gamma ,\lambda }\to{\mathbb{Z}}_{\ell}^{\times }$ denotes the canonical $\ell $-orientation on $G_{\Gamma ,\lambda }$. This yields a plethora of new examples of pro-$\ell $ groups that are not maximal pro-$\ell $ Galois groups. 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, for example, 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 $\operatorname{Im}(\lambda )\subseteq 1+4{\mathbb{Z}}_{2}$ in case that $\ell =2$, then $H^{\bullet }(G_{\Gamma ,\lambda },{\mathbb{F}}_{\ell }) \simeq \Lambda ^{\bullet }(\ddot{\Gamma }^{\textrm{op}})$ generalizing a well-known result of M. Salvetti (cf. [ 39]). Dedicated to the memory of Avinoam Mann.