Right-angled Artin groups and enhanced Koszul properties

Abstract

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

To appear in J. Group Theory