Oriented pro-\(\ell\) groups with the Bogomolov-Positselski property (Q2122581)

By an \(\ell\)-oriented profinite group, for a prime number \(\ell\), we mean a profinite group \(G\) together with a continuous homomorphism of profinite groups \(\theta \colon G \to \mathbb{Z}_{\ell}^{\ast}\), where \(\mathbb{Z}_{\ell}^{\ast}\) denotes the group of units of the ring \(\mathbb{Z}_{\ell}\) of \(\ell\)-adic integers. When \((G, \theta)\) is an \(\ell\)-oriented pro-\(\ell\)-group, it is simply called an oriented pro-\(\ell\) group. Apart from \(\operatorname{ker}(\theta)\), an oriented pro-\(\ell\) group \((G, \theta)\) contains two distinguished closed subgroups, denoted by \(K_{\theta}(G)\) and \(I_{\theta}(G)\), and defined as follows: \(K_{\theta}(G) = cl(\langle h^{-\theta (g)}ghg^{-1} \mid g \in G, h \in\operatorname{ker}(\theta)\rangle)\); \(I_{\theta}(G) = cl(\langle h \in \operatorname{ker}(\theta) \mid \exists k \in \mathbb{N}_0\colon h^{\ell ^{k}} \in K_{\theta}(G)\rangle)\). We say that \((G, \theta)\) is \(\theta\)-abelian if \(K_{\theta}(G)\) is trivial and \(\operatorname{ker}(\theta)\) is a free abelian pro-\(\ell\) group (in this case, it turns out that \(G\) is a free abelian-by-cyclic pro-\(\ell\) group, for \(\ell \neq 2\)). Generally, \(K_{\theta}(G)\) is a closed normal subgroup of \(G\) included in the Frattini subgroup \(\Phi (G) = cl(G^{\ell}.[G, G])\) of \(G\), and such that \([\operatorname{ker}(\theta), \operatorname{ker}(\theta)] \subseteq K_{\theta}(G) \subseteq \operatorname{ker}(\theta)\); in particular, the quotient \(\operatorname{ker}(\theta)/K_{\theta}(G)\) is an abelian pro-\(\ell\) group, and \(I_{\theta}(G)/K_{\theta}(G)\) is its torsion subgroup. The group \(G(\theta) = G/I_{\theta}(G)\), considered with the homomorphism \(\tilde \theta\) induced by \(\theta\), is a maximal \(\theta\)-abelian quotient of \((G, \theta)\). By definition, the oriented pro-\(\ell\) group \((G, \theta)\) has the Bogomolov-Positselski property if the kernel of the canonical projection \(\pi_{G,\theta}^{ab}\colon G \to G(\theta)\) is a free pro-\(\ell\)-group contained in the Frattini subgroup of \(G\). Given a field \(\mathbb{K}\) with a separable closure \(\mathbb{K}^{sep}\), the absolute Galois group \(\mathcal{G}_{\mathbb{K}} := \mathcal{G}(\mathbb{K}^{sep}/\mathbb{K})\) carries a natural cyclotomic \(\ell\)-orientation \(\tilde \theta_{\mathbb{K},\ell}\colon \mathcal{G}_{\mathbb{K}} \to \mathbb{Z}_{\ell}^{\ast}\) (see [\textit{C. Quadrelli} and \textit{T. S. Weigel}, Doc. Math. 25, 1881--1916 (2020; Zbl 1467.12009)]). The paper under review shows that every oriented pro-\(\ell\)-group of elementary type, in the sense of Efrat, has the Bogomolov-Positselski property. In addition, it is shown that, for an \(H^{\ast}\)-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. As an application of the former theorem, the authors prove that Efrat's Elementary Type Conjecture implies a positive answer to Positselski's conjecture on the maximal pro-\(\ell\) Galois group of \(\mathbb{K}\) in the special case where the quotient group \(\mathbb{K}^{\ast}/\mathbb{K}^{\ast \ell}\) is finite. Positselski's conjecture has been motivated by an earlier conjecture formulated by \textit{F. A. Bogomolov} [Proc. Symp. Pure Math. 58, 83--88 (1995; Zbl 0843.12003)], and \textit{L. Positselski} [Int. Math. Res. Not. 2005, No. 31, 1901--1936 (2005; Zbl 1160.19301)]. It states that if \(\mathbb{K}\) contains a primitive \(2\ell\)-th root of unity and \(\mathbb{K}^{\prime}\) is the extension of \(\mathbb{K}\) in \(\mathbb{K}^{sep}\) obtained by adjunction of the \(\ell^n\)-th roots of the elements of \(\mathbb{K}\), for each \(n \in \mathbb{N}\), then the maximal pro-\(\ell\) Galois group of \(\mathbb{K}^{\prime}\) is a free pro-\(\ell\) group.