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

Abstract

For a prime number $p$ and an integer $m\ge 2$, 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 ${\rho }^{\ast }(\bar{\omega })$ of a given cohomology element $\bar{\omega }\in H^n(\bar{G},{\mathbb{F}}_p)$, where $\bar{G}$ is a finite $p$-group, $n\ge 2$, and $\rho :G\to \bar{G}$ is a pro-$p$ group homomorphism.

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