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 $H\mapsto \Phi \left(H\right)$, from the poset of all closed topologically 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 finite 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 $Z/p$-cohomology of a Frattini-resistant pro-p group there are no essential triple Massey products.