Oriented right-angled Artin pro-ℓ groups and maximal pro-ℓ Galois groups

For a prime number ℓ we introduce and study oriented right-angled Artin pro-ℓ groups GΓ,λ(oriented pro-ℓ RAAGs for short) associated to a finite oriented graph Γ and a continuous group homomorphism λ: Zℓ → Z×ℓ . We show (cf. Thm. 1.1) that an oriented pro-ℓ RAAG GΓ,λ is a Bloch-Kato pro-ℓ group if, and only if, (GΓ,λ, θΓ,λ) is an oriented pro-ℓ group of elementary type generalizing a recent result of I. Snopche and P. Zalesskiĭ (cf. [44]). Here θΓ,λ : GΓ,λ → Z×p denotes the canonical ℓ-orientation on GΓ,λ. We invest some effort in order to show that oriented right-angled Artin pro-ℓ groups share many properties with right-angled Artin pro-ℓ-groups or even discrete RAAG’s, e.g., if Γ is a specially oriented chordal graph, then GΓ,λ is coherent (cf. Thm. 1.3(ii)) generalizing a result of C. Droms (cf. [10]) . Moreover, in this case (GΓ,λ, θΓ,λ) has the Positselski-Bogomolov property (cf. Thm. 1.3(i)) generalizing a result of H. Servatius, C. Droms and B. Servatius for discrete RAAG’s (cf. [43]). If Γ is a specially oriented chordal graph and Im(λ) ⊆ 1+4Z2 in case that ℓ = 2, then H•(GΓ,λ, Fℓ) ≃ Λ•(Γ̈op) (cf. Thm. 1.3(iii)) generalizing a well known result of M. Salvetti (cf. [40]).