Groups of p-absolute Galois type that are not absolute Galois groups

TL;DR

This paper identifies new classes of pro-p groups of p-absolute Galois type, including certain right-angled Artin groups and Demushkin groups, which satisfy the Massey vanishing property but are not actual absolute Galois groups.

Contribution

It proves that specific pro-p groups, such as those from chordal graphs and Demushkin groups, are of p-absolute Galois type and satisfy the Massey vanishing property, expanding known examples.

Findings

Pro-p completion of right-angled Artin groups of chordal graphs are of p-absolute Galois type.

Demushkin groups are of p-absolute Galois type.

Certain group constructions preserve the p-absolute Galois type and Massey vanishing property.

Abstract

Let p be a prime. We study pro-p groups of p-absolute Galois type, as defined by Lam-Liu-Sharifi-Wake-Wang. We prove that the pro-p completion of the right-angled Artin group associated to a chordal simplicial graph is of p-absolute Galois type, and moreover it satisfies a strong version of the Massey vanishing property. Also, we prove that Demushkin groups are of p-absolute Galois type, and that the free pro-p product -- and, under certain conditions, the direct product -- of two pro-p groups of p-absolute Galois type satisfying the Massey vanishing property is again a pro-p group of p-absolute Galois type satisfying the Massey vanishing property. Consequently, there is a plethora of pro-p groups of p-absolute Galois type satisfying the Massey vanishing property that do not occur as absolute Galois groups.