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

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.

Massey products in Galois cohomology and the Elementary Type Conjecture

“…The author wishes to thank I. Efrat, E. Matzri, J. Minac, F.W. Pasini, and N.D. Tân, for several inspiring discussions on Massey products in Galois cohomology and pro-p groups of elementary type, which occurred in the past years; and S. Blumer and A. Cassella for working together on the paper [2], where the strong Massey vanishing property is defined. Also, special thanks are due to A. Ioppolo and A. Previtali, who generously provided the author enough time to complete this paper.…”