Chasing maximal pro-p Galois groups via 1-cyclotomicity

Abstract

Let $p$ be a prime. We prove 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, and thus do not occur as maximal pro-$p$ Galois groups of fields containing a root of 1 of order $p$. We show that other cohomological obstructions which are used to detect pro-$p$ groups that are not maximal pro-$p$ Galois groups - the quadraticity of $\mathbb{Z}/p\mathbb{Z}$-cohomology and the vanishing of Massey products - fail with the above pro-$p$ groups. Finally, we prove that the Minač-Tân pro-$p$ group cannot give rise to a 1-cyclotomic oriented pro-$p$ group, and we conjecture that every 1-cyclotomic oriented pro-$p$ group satisfy the strong $n$-Massey vanishing property for $n>2$.

Thorough revision of the old version (v3), revision of the previous version following the referee's comments (v4-v6). Conjecture 1.5 (relating Massey products and 1-cyclotomicity) has been extended to all n>2

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Cited by in corpus (1)

• Directed graphs, Frattini-resistance, and maximal pro-$p$ Galois groups