Higher Massey Products in Demuškin Variations: Support Blocks, One-Relator Reduction, and a Five-Fold Vanishing Case
Abstract
Blumer and Quadrelli introduced a family ℱ2 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 vh=(αh(z1),αh(z2))∈Fp2 of a defined n-fold Massey product. Definability forces det(vh,vh+1)=0, so the nonzero profile entries decompose into support blocks carrying projective directions in PP1(𝔽p). After the known endpoint reduction, profiles with exactly r blocks are counted by binomn−12r, 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 x1q for every parameter allowed in ℱ2. In length five, this yields vanishing whenever all three interior profile vectors v2,v3,v4 are nonzero, for every prime p. The resulting classification identifies the remaining five-fold support types and the compatibility mechanisms governing them.
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BibTeX
@misc{palaisti2026higher,
title = {{Higher Massey Products in Demuškin Variations: Support Blocks, One-Relator Reduction, and a Five-Fold Vanishing Case}},
author = {Marina Palaisti},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00253},
eprint = {2609.00253},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00253},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Higher Massey Products in Demuškin Variations: Support Blocks, One-Relator Reduction, and a Five-Fold Vanishing Case
AU - Marina Palaisti
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00253
UR - https://arxiv.org/abs/2609.00253
AB - Blumer and Quadrelli introduced a family 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))∈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(𝔽_p). After the known endpoint reduction, profiles with exactly r blocks are counted by binomn−12r, 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 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.
ER -