Groups of p-absolute Galois type that are not absolute Galois groups

Demushkin groups of uncountable rank

“…Denote by 𝐺 𝐹 its absolute Galois group, and by 𝐺 𝐹 (𝑝) its maximal pro-𝑝 quotient. Notice that 𝐺 𝐹 (𝑝) is isomorphic to Gal(𝐹(𝑝)∕𝐹) where 𝐹(𝑝) denotes the maximal 𝑝-extension of 𝐹, that is, the compositum of all its finite 𝑝-extension, and hence it is sometimes called "The maximal pro-𝑝 Galois group of 𝐹" (see, e.g., [2]).…”

“…Denote by $$G_F$$ its absolute Galois group, and by $$G_F(p)$$ its maximal pro‐$$p$$ quotient. Notice that $$G_F(p)$$ is isomorphic to $$\operatorname{Gal}(F(p)/F)$$ where $$F(p)$$ denotes the maximal $$p$$‐extension of $$F$$, that is, the compositum of all its finite $$p$$‐extension, and hence it is sometimes called “The maximal pro‐$$p$$ Galois group of $$F$$” (see, e.g., [2]).…”

“…Let 𝑞 ≠ 2 be a prime power or equal to 0, and 𝜇 > ℵ 0 . For every nondegenerate skew symmetric bilinear form (𝑉, 𝜑) of dimension 𝑉 there is a Demushkin group 𝐺 of rank 𝜇, with 𝑞(𝐺) = 𝑞, 𝑠(𝐺) = 0, and whose cup-product bilinear form 𝐻 1 (𝐺) ∪ 𝐻 1 (𝐺) → 𝐻 2 (𝐺) is isomorphic to (𝑉, 𝜑). Corollary 2.…”

Chasing maximal pro-p Galois groups via 1-cyclotomicity

“…In fact, by [21, Thm. 1.1], there are many more oriented graphs whose associated oriented pro-p RAAGs yield no essential n-fold Massey products, than oriented graphs yielding Frattini-resistant pro-p RAAGs: for example, for every unoriented graphand thus also the unoriented graphs (4.1), which are not of elementary type -the associated oriented pro-p RAAGs yield no essential n-fold Massey products for any n > 2 (see also [2,Thm. 1.1]).…”