A Bloch-Kato pro-p group G is a pro-p group with the property that the Fp-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 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/ D cd.G/, and its Fp-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.

Bloch-Kato pro-p groups and locally powerful groups

Quadrelli, C
2014-01-01

Abstract

A Bloch-Kato pro-p group G is a pro-p group with the property that the Fp-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 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/ D cd.G/, and its Fp-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.
2014
2014
Galois cohomology; Bloch-Kato groups; powerful pro-p groups; Tits alternative; elementary type conjecture
Quadrelli, C
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2129366
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