Variations of Demushkin Groups that are not Absolute Galois Groups

TL;DR

The paper constructs new pro-p groups with elementary presentations that cannot be realized as maximal pro-p Galois groups of fields containing a p-th root of unity, challenging existing Galois group characterizations.

Contribution

It introduces two families of pro-p groups that do not fit into known Galois group frameworks, expanding the understanding of possible Galois group structures.

Findings

These pro-p groups cannot be completed into 1-cyclotomic oriented pro-p groups.

They do not satisfy the triple Massey vanishing property.

They are not ruled out by known cohomological properties of Galois groups.

Abstract

We construct two families of examples of pro-p groups, with rather elementary presentations, that do not complete into 1-cyclotomic oriented pro-p groups. These provide brand new examples of pro-p groups that do not occur as maximal pro-p Galois groups of fields containing a root of unity of order p - and thus, as absolute Galois groups. Moreover, we show that these pro-p groups may not be ruled out as maximal pro-p Galois groups employing other cohomological properties that are known to hold for all maximal pro-p Galois groups, such as the triple Massey vanishing property, or the quadraticity of Fp-cohomology.