Massey products in Galois cohomology and the Elementary Type Conjecture

Abstract

Let $p$ be a prime. We prove that a positive solution to Efrat's Elementary Type Conjecture implies a positive solution to the strengthened version of Mina\v{c}--T\^an's Massey Vanishing Conjecture in the case of finitely generated maximal pro-$p$ Galois groups whose pro-$p$ cyclotomic character has torsion-free image. Consequently, the maximal pro-$p$ Galois group of a field $\mathbb{K}$ containing a root of 1 of order $p$ (and also \sqrt{-1} if $p=2$) satisfies the strong $n$-Massey vanishing property for every $n>2$ (which is equivalent to the cup-defining $n$-Massey product property for every $n>2$, as defined by Mina\v{c}--T\^an) in several relevant cases.