Bloch-Kato pro-p groups and locally powerful groups

TL;DR

This paper characterizes Bloch-Kato pro-p groups, showing they are either free of infinite rank or theta-abelian, and explores their properties and implications for Galois groups and the Elementary type conjecture.

Contribution

It establishes a dichotomy for Bloch-Kato pro-p groups and links their structure to powerful, p-adic analytic groups, advancing understanding of their Galois-theoretic significance.

Findings

Either no closed free pro-p groups of infinite rank are contained

Finitely generated groups are powerful and p-adic analytic

Their cohomology rings are exterior algebras

Abstract

A Bloch-Kato pro-p group G is a pro-p group with the property that the 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 $\theta\colon G\rightarrow \Z_p^\times$ such that G is theta-abelian. In case that G is also finitely generated, this implies that G is powerful, p-adic analytic with d(G)=cd(G), and its \F_p-cohomology ring is an exterior algebra. These results will be obtained by studying locally powerful groups (see Theorem A). 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 (see Corollary 4.9). Finally, we study certain closure operations of the class of Bloch-Kato pro-p groups, connected with the Elementary type conjecture.