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

Abstract

For a prime number $\ell$ we say that an oriented pro-$\ell$ group $(G,\theta )$ has the Bogomolov-Positselski property if the kernel of the canonical projection on its maximal $\theta$-abelian quotient ${\pi }_{G,\theta }^{ab}:G\to G(\theta )$ is a free pro-$\ell$ group contained in the Frattini subgroup of $G$. We show that oriented pro-$\ell$ groups of elementary type have the Bogomolov-Positselski property. This shows that Efrat's Elementary Type Conjecture implies a positive answer to Positselski's version of Bogomolov's Conjecture on maximal pro-$\ell$ Galois groups of a field $K$ in case that $K^{\times }/(K^{\times }{)}^{\ell }$ is finite. Secondly, it is shown that for an $H^{\bullet }$-quadratic oriented pro-$\ell$ group $(G,\theta )$ the Bogomolov-Positselski property can be expressed by the injectivity of the transgression map $d_2^{2,1}$ in the Hochschild-Serre spectral sequence.

The title of the old version of this paper was «Oriented pro-$\ell$ groups with the Bogomolov property»: following te referee's advice, we changed the name of the property we study from «Bogomolov property» to «Bogomolov-Positselski property»

References in corpus (3)

• Koszul property and Bogomolov's conjecture
• Bloch-Kato pro-p groups and locally powerful groups
• Approximation in $L^p(\mu )$ with deep ReLU neural networks

Cited by in corpus (4)

• Galois-theoretic features for 1-smooth pro-$p$ groups
• Massey products in Galois cohomology and the Elementary Type Conjecture
• Oriented right-angled Artin pro-$\ell$ groups and maximal pro-$\ell$ Galois groups
• Chasing maximal pro-p Galois groups via 1-cyclotomicity