Oriented pro-$\ell$ groups with the Bogomolov-Positselski property

Abstract

For a prime number ℓ we say that an oriented pro-ℓ group (G, θ) has the Bogomolov property if the kernel of the canonical projection on its maximal θ-abelian quotient π ab G,θ : G → G(θ) is a free pro-ℓ group contained in the Frattini subgroup of G. We show that pro-ℓ groups of elementary type have the Bogomolov property (cf. Theorem 1.2). This shows that Efrat's Elementary Type Conjecture implies a positive answer to Positselski's version of Bogomolov's Conjecture on maximal pro-ℓ Galois groups of a field K in case that K × /(K × ) ℓ is finite. Secondly, it is shown that for an H • -quadratic oriented pro-ℓ group (G, θ) the Bogomolov property can be expressed by the injectivity of the transgression map d 2,1 2 in the Hochschild-Serre spectral sequence (cf. Theorem 1.4).Then the maximal pro-ℓ Galois group of ℓ ∞ √ K is a free pro-ℓ group.A profinite group G admits a maximal pro-ℓ quotient G(ℓ) = G/O ℓ (G), where O ℓ (G) is the closed normal subgroup of G being generated by all pro-q Sylow

Galois-theoretic features for 1-smooth pro-p groups

“…From [9, Theorem 5.6], one may deduce also the following grouptheoretic characterization of Kummerian pro-p pairs: a pro-p group G may complete into a Kummerian oriented pro-p group if, and only if, there exists an epimorphism of pro-p groups φ∶ G ↠ Ḡ such that Ḡ has a minimal presentation (2.5), and Ker(φ) is contained in the Frattini subgroup of G (cf., e.g., [22,Proposition 3.11]).…”

Chasing maximal pro-p Galois groups via 1-cyclotomicity

“…Subsequently, we decided to change strategy, and to split the original paper: this manuscript is the evolution of one of the two resulting pieces. In the meanwhile, the research on 1-cyclotomic oriented pro-p groups went on, and Theorems 1.1-1.2 have been used for proving results in [3,29,32].…”