Higher Massey Products in Demu\v{s}kin 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 Demu\v{s}kin group by imposing the commutativity of two generators which are not paired in the Demu\v{s}kin relation. They proved the triple and quadruple Massey vanishing properties and asked whether the same holds in every length. We introduce the $z$-profile \(v_h=(\alpha_h(z_1),\alpha_h(z_2))\in\Fp^2\) 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 Demu\v{s}kin 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.