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

TL;DR

This paper extends Labute's theory of mild pro-p-groups to higher Massey products, enabling the construction of groups with relations of arbitrary degree and providing new insights into their cohomological properties.

Contribution

It generalizes the theory of mild pro-p-groups to higher Massey products and introduces methods to construct groups with complex defining relations.

Findings

Extended Labute's theory to weighted Zassenhaus filtrations

Proved a generalization for higher Massey products

Provided new evidence related to Serre's open question

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 cd G=2 if $H^1(G,\F_p)=U\oplus V$ as $\F_p$-vector space and the cup-product $H^1(G,\F_p)\otimes H^1(G,\F_p)\to H^2(G,\F_p)$ maps $U\otimes V$ surjectively onto $H^2(G,\F_p)$ and is identically zero on $V\otimes V$. 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 for one-relator pro-p-groups and obtain some new evidence of an open question due to Serre.