On pro-$p$ groups with quadratic cohomology

Abstract

The main purpose of this article is to study pro-$p$ groups with quadratic ${\mathbb{F}}_p$-cohomology algebra, i.e. $H^{\bullet }$-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 $H^{\bullet }$-quadratic pro-$p$ groups is $H^{\bullet }$-quadratic, under certain necessary conditions. Moreover, we introduce and investigate a new family of pro-$p$ groups that yields many new examples of $H^{\bullet }$-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 $H^{\bullet }$-quadratic pro-$p$ groups.

40 pages

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Cited by in corpus (9)

• Groups of p-absolute Galois type that are not absolute Galois groups
• Pro-p groups with few relations and Universal Koszulity
• Right-angled Artin pro-$p$ groups
• Oriented right-angled Artin pro-$\ell$ groups and maximal pro-$\ell$ Galois groups
• One relator maximal pro-p Galois groups and the Koszulity conjectures
• Chasing maximal pro-p Galois groups via 1-cyclotomicity
• Frattini-injectivity and Maximal pro-$p$ Galois groups
• Directed graphs, Frattini-resistance, and maximal pro-$p$ Galois groups
• Generalized Stallings' decomposition theorem for pro-$p$ groups