Right‐angled Artin pro‐p$p$ groups

Abstract

Let p $p$ be a prime. The right-angled Artin pro- p $p$ group G Γ $G_{\Gamma }$ associated to a finite simplicial graph Γ $\Gamma$ is the pro- p $p$ completion of the right-angled Artin group associated to Γ $\Gamma$ . We prove that the following assertions are equivalent: (i) no induced subgraph of Γ $\Gamma$ is a square or a line with four vertices (a path of length 3); (ii) every closed subgroup of G Γ $G_{\Gamma }$ is itself a right-angled Artin pro- p $p$ group (possibly infinitely generated); (iii) G Γ $G_{\Gamma }$ is a Bloch–Kato pro- p $p$ group; (iv) every closed subgroup of G Γ $G_{\Gamma }$ has torsion free Abelianization; (v) G Γ $G_{\Gamma }$ occurs as the maximal pro- p $p$ Galois group G K ( p ) $G_K(p)$ of some field K $K$ containing a primitive p $p$ th root of unity; (vi) G Γ $G_{\Gamma }$ can be constructed from Z p $\mathbb {Z}_p$ by iterating two group theoretic operations, namely, direct products with Z p $\mathbb {Z}_p$ and free pro- p $p$ products. This settles in the affirmative a conjecture of Quadrelli and Weigel. Also, we show that the Smoothness Conjecture of De Clercq and Florence holds for right-angled Artin pro- p $p$ groups. Moreover, we prove that G Γ $G_{\Gamma }$ is coherent if and only if each circuit of Γ $\Gamma$ of length greater than three has a chord.