Two families of pro-p groups that are not absolute Galois groups

TL;DR

This paper introduces two new families of pro-p groups that cannot be realized as absolute Galois groups, using properties like 1-smoothness, quadraticity of Galois cohomology, and Massey product vanishing.

Contribution

It identifies specific pro-p groups that are not absolute Galois groups, expanding understanding of Galois group obstructions.

Findings

Two new families of pro-p groups are not realizable as Galois groups.

One family includes one-relator pro-p groups potentially not Galois groups.

Uses properties like 1-smoothness and cohomological conditions to establish non-realizability.

Abstract

Let $p$ be a prime. We produce two new families of pro-$p$ groups which are not realizable as absolute Galois groups of fields. To prove this we use the 1-smoothness property of absolute Galois pro-$p$ groups. Moreover, we show in these families one has one-relator pro-$p$ groups which may not be ruled out as absolute Galois groups employing the quadraticity of Galois cohomology (a consequence of Rost-Voevodsky Theorem), or the vanishing of Massey products in Galois cohomology.