Right-angled Artin pro-$p$ groups

Abstract

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