Chasing maximal pro-\(p\) Galois groups via 1-cyclotomicity (Q6537093)

Let \(p\) be a prime number, and let \(1+p\mathbb{Z} \sb p\) denote the pro-\(p\) group of principal units of the ring \(\mathbb{Z} \sb p\) of \(p\)-adic integers, namely, \(1+p\mathbb{Z} \sb p = \{1+p\lambda \colon \lambda \in \mathbb{Z} \sb p\}\). An oriented pro-\(p\) group is a pair \((G, \theta )\) consisting of a pro-\(p\) group \(G\) and a morphism of pro-\(p\) groups \(\theta : G \to 1 + p\mathbb{Z}p\), called an orientation of \(G\) (see [\textit{C. Quadrelli} and \textit{T. S. Weigel}, Doc. Math. 25, 1881--1916 (2020; Zbl 1467.12009)]; oriented pro-\(p\) groups were introduced by Efrat uder the name of cyclotomic pro-\(p\) pairs (see [\textit{I. Efrat}, Manuscr. Math. 95, No. 2, 237--249 (1998; Zbl 0902.12003)]). An oriented pro-p group \((G, \theta )\) gives rise to the continuous \(G\)-module \(\mathbb{Z}\sb p(\theta )\), which is equal to \(\mathbb{Z}\sb p\) as an abelian pro-\(p\) group, and which is endowed with the continuous \(G\)-action defined by \(g.\lambda = \theta (g).\lambda \), for all \(g \in G\), \(\lambda \in \mathbb{Z}\sb p(\theta )\). An oriented pro-\(p\) group \((G, \theta )\) is called Kummerian, provided that for every \(n \ge 1\) the natural morphism \(H\sp 1(G, \mathbb{Z}\sb p(\theta )/p \sp n\mathbb{Z}\sb p(\theta )) \to H\sp 1(G, \mathbb{Z}/p\mathbb{Z}), \ (1.1)\) induced by the epimorphism of continuous \(G\)-modules \(\mathbb{Z}\sb p(\theta )/p \sp n\mathbb{Z}\sb p(\theta ) \to \mathbb{Z}/p\) is surjective, where \(\mathbb{Z}/p\) is viewed as a trivial \(G\)-module (see [\textit{I. Efrat} and \textit{C. Quadrelli}, J. Algebra 525, 284--310 (2019; Zbl 1446.12005)]). Moreover, \((G, \theta )\) is called \(1\)-cyclotomic (the same as \(1\)-smooth in the author's papers [\textit{C. Quadrelli}, Can. Math. Bull. 65, No. 2, 525--541 (2022; Zbl 1514.12003); Homology Homotopy Appl. 24, No. 2, 53--67 (2022; Zbl 1512.12003)] if the natural morphism (1.1) is surjective also with \(H\) instead of \(G\), and with the restriction \(\theta \vert H\colon H \to 1+p\mathbb{Z}\sb p\) instead of \(\theta \), for all closed subgroups \(H\) of \(G\). This cohomological condition has first been considered by Labute, who has shown ante litteram that for every Demushkin group \(G\) there exists precisely one orientation which completes \(G\) into a Kummerian oriented pro-\(p\) group, namely the orientation induced by the dualizing module of \(G\) (see [\textit{J. P. Labute}, Can. J. Math. 19, 106--132 (1967; Zbl 0153.04202)]).\N\NThe paper under review proves that certain amalgamated free pro-\(p\) products of Demushkin groups with pro-\(p\)-cyclic amalgam cannot give rise to a \(1\)-cyclotomic oriented pro-\(p\) group, whence, they are not realizable as maximal pro-\(p\) Galois groups of fields containing a primitive \(p\)-th root of \(1\) (this implies that the considered groups are not realizable as absolute Galois groups). The paper gives a pro-\(p\) group presentation of these groups by generators and defining relations. It shows that other cohomological obstructions which have successfully been used to detect pro-\(p\) groups that are not maximal pro-\(p\) Galois groups, such as the quadraticity of \(\mathbb{Z}/p\mathbb{Z}\)-cohomology and the vanishing of Massey products, fail with the above pro-\(p\) groups. Also, the author proves that the Mináč-Tân pro-\(p\) group cannot give rise to a \(1\)-cyclotomic oriented pro-\(p\) group, and he conjectures that every \(1\)-cyclotomic oriented pro-\(p\) group satisfies the strong \(n\)-Massey vanishing property for \(n = 3, 4\).