Higher Massey Products in Demuškin Variations: Support Blocks, One-Relator Reduction, and a Five-Fold Vanishing Case

Abstract

Blumer and Quadrelli introduced a family $\mathcal{F}_2$ of two-relator pro-$p$ groups obtained from a Demuškin group by imposing the commutativity of two generators which are not paired in the Demuškin relation. They proved the triple and quadruple Massey vanishing properties and asked whether the same holds in every length. We introduce the $z$-profile \(v_h=(α_h(z_1),α_h(z_2))\in\Fp^2\) of a defined $n$-fold Massey product. Definability forces $\det(v_h,v_{h+1})=0$, so the nonzero profile entries decompose into support blocks carrying projective directions in $\PP^1(\mathbb{F}_p)$. After the known endpoint reduction, profiles with exactly $r$ blocks are counted by $\binom{n-1}{2r}$, and $r$ blocks first occur in length $2r+1$. We also prove an all-length conditional reduction for Dwyer's lifting problem: if the added commuting relator can be made exact in a lift, then the remaining central defect of the Demuškin relator can be removed by an endpoint correction. The correction preserves the power term $x_1^q$ for every parameter allowed in $\mathcal{F}_2$. In length five, this yields vanishing whenever all three interior profile vectors $v_2,v_3,v_4$ are nonzero, for every prime $p$. The resulting classification identifies the remaining five-fold support types and the compatibility mechanisms governing them.