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

TL;DR

This paper proves that certain oriented pro-ll groups, specifically those of elementary type, possess the Bogomolov-Positselski property, linking group-theoretic structures to conjectures in Galois theory.

Contribution

It demonstrates that oriented pro-ll groups of elementary type have the Bogomolov-Positselski property, supporting conjectures relating Galois groups and field properties.

Findings

Elementary type groups have the Bogomolov-Positselski property.

The property relates to the kernel being a free pro-ll group.

Injectivity of a Hochschild-Serre spectral sequence map characterizes the 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 $\pi^{ab}_{G,\theta}\colon G\to G(\theta)$ 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.