Higher Massey products in the cohomology of mild pro-p-groups

Abstract

Translating results due to J. Labute into group cohomological language, A. Schmidt proved that a finitely presented pro-p-group G is mild and hence of cohomological dimension cdG=2 if H1(G,Fp)=U⊕V as Fp-vector space and the cup-product H1(G,Fp)⊗H1(G,Fp)→H2(G,Fp) maps U⊗V surjectively onto H2(G,Fp) and is identically zero on V⊗V. This has led to important results in the study of p-extensions of number fields with restricted ramification, in particular in the case of tame ramification. In this paper, we extend Labute's theory of mild pro-p-groups with respect to weighted Zassenhaus filtrations and prove a generalization of the above result for higher Massey products which allows to construct mild pro-p-groups with defining relations of arbitrary degree. We apply these results to one-relator pro-p-groups and obtain some new evidence of an open question due to Serre.