On pro-p groups with quadratic cohomology

The main purpose of this article is to study pro-p groups with quadratic Fp-cohomology algebra, i.e. H•-quadratic pro-p groups. Prime examples of such groups are the maximal Galois pro-p groups of fields containing a primitive root of unity of order p. We show that the amalgamated free product and HNN-extension of H•-quadratic pro-p groups is H•-quadratic, under certain necessary conditions. Moreover, we introduce and investigate a new family of pro-p groups that yields many new examples of H•-quadratic groups: p-RAAGs. These examples generalise right angled Artin groups in the category of pro-p groups. Finally, we explore “Tits alternative behaviour” of H•-quadratic pro-p groups.