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

For a prime number , we introduce and study oriented right-angled Artin pro- groups G,λ(oriented pro- RAAGs for short) associated to a finite oriented graph and a continuous group homomorphism λ: Z → Z× . We show that an oriented pro- RAAG G,λ is a Bloch–Kato pro- group if, and only if, (G,λ, θ,λ) is an oriented pro- group of elementary type, generalizing a recent result of I. Snopce and P. Zalesski—here θ,λ : G,λ → Z× denotes the canonical l-orientation on G,λ. This yields a plethora of new examples of pro-groups that are not maximal pro-l Galois groups. We invest some effort in order to show that oriented right-angled Artin pro- groups share many properties with right-angled Artin pro-l-groups or even discrete RAAG’s, for example, if is a specially oriented chordal graph, then G,λ is coherent generalizing a result of C. Droms. Moreover, in this case, (Gλ, θλ) has the Positselski–Bogomolov property generalizing a result of H. Servatius, C. Droms, and B. Servatius for discrete RAAG’s. If is a specially oriented chordal graph and Im(λ) ⊆ 1 + 4Z2 in case that = 2, then H•(G,λ, F) •(̈op) generalizing a well-known result of M. Salvetti (cf. [39]). Dedicated to the memory of Avinoam Mann.