Let ๐ฝ be a finite field.
We prove that the cohomology algebra H^{\bullet}(G_{\Gamma},\mathbb{F}) with coefficients in ๐ฝ of a right-angled Artin group G_{\Gamma} is a strongly Koszul algebra for every finite graph ฮ.
Moreover, H^{\bullet}(G_{\Gamma},\mathbb{F}) is a universally Koszul algebra if, and only if, the graph ฮ associated to the group G_{\Gamma} has the diagonal property.
From this, we obtain several new examples of pro-๐ groups, for a prime number ๐, whose continuous cochain cohomology algebra with coefficients in the field of ๐ elements is strongly and universally (or strongly and non-universally) Koszul.
This provides new support to a conjecture on Galois cohomology of maximal pro-๐ Galois groups of fields formulated by J. Minรกฤ et al.
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.
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