The symbol length for elementary type pro-$p$ groups and Massey products

Abstract

For a prime number $p$ and an integer $m\geq2$, we prove that the symbol length of all elements of $m$-fold Massey products in $H^2(G,\mathbb{F}_p)$, for pro-$p$ groups $G$ of elementary type, is bounded by $(m^2/4)+m$. Assuming the Elementary Type Conjecture, this applies to all finitely generated maximal pro-$p$ Galois groups $G=G_F(p)$ of fields $F$ which contain a root of unity of order $p$. More generally, we provide such a uniform bound for the symbol length of all pullbacks $\rho^*(\bar\omega)$ of a given cohomology element $\bar\omega\in H^n(\bar G,\mathbb{F}_p)$, where $\bar G$ is a finite $p$-group, $n\geq2$, and $\rho\colon G\to \bar G$ is a pro-$p$ group homomorphism.