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

For a prime number $\ell$ we say that an oriented pro-$\ell$ group $(G,θ)$ has the Bogomolov-Positselski property if the kernel of the canonical projection on its maximal $θ$-abelian quotient $π^{ab}_{G,θ}\colon G\to G(θ)$ 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 $(G,θ)$ the Bogomolov-Positselski property can be expressed by the injectivity of the transgression map $d_2^{2,1}$ in the Hochschild-Serre spectral sequence.

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