Bloch–Kato pro-p groups and locally powerful groups

Abstract

Abstract. A Bloch–Kato pro-p group G is a pro-p group with the property that the 𝔽 p $\mathbb {F}_p$ -cohomology ring of every closed subgroup of G is quadratic. It is shown that either such a pro-p group G contains no closed free pro-p groups of infinite rank, or there exists an orientation θ : G → ℤ p × $\theta \colon G\rightarrow \mathbb {Z}_p^\times $ such that G is θ-abelian. In case that G is also finitely generated, this implies that G is powerful, p-adic analytic with d ( G ) = cd ( G ) $d(G)=\operatorname{cd}(G)$ , and its 𝔽 p $\mathbb {F}_p$ -cohomology ring is an exterior algebra. These results will be obtained by studying locally powerful groups. There are certain Galois-theoretical implications, since Bloch–Kato pro-p groups arise naturally as maximal pro-p quotients and pro-p Sylow subgroups of absolute Galois groups. Finally, we study certain closure operations of the class of Bloch–Kato pro-p groups, connected with the Elementary Type Conjecture.