Directed graphs, Frattini-resistance, and maximal pro-$p$ Galois groups

Abstract

Let $p$ be a prime. Following Snopce-Tanushevski, a pro-$p$ group $G$ is called Frattini-resistant if the function $H\mapsto\Phi(H)$, from the poset of all closed finitely-generated subgroups of $G$ into itself, is a poset embedding. We prove that for an oriented right-angled Artin pro-$p$ group (oriented pro-$p$ RAAG) $G$ associated to a directed graph the following four conditions are equivalent: the associated directed graph is of elementary type; $G$ is Frattini-resistant; every topologically finitely generated closed subgroup of $G$ is an oriented pro-$p$ RAAG; $G$ is the maximal pro-$p$ Galois group of a field containing a root of 1 of order $p$. Also, we conjecture that in the $\mathbb{Z}/p$-cohomology of a Frattini-resistant pro-$p$ group there are no essential triple Massey products.