Detecting Cohomological Dimension Three in Free-by-Demuškin Pro-p Groups
Abstract
Let 1⟶ N⟶ G⟶ D⟶1 be exact, with N≠1 free pro-p of arbitrary rank and D Demuškin. Writing W=N/Φ(N), we prove H3(G,𝔽p)∨≃ WD, cdp G=3 if and only if WD≠0. Thus dimension three is detected on the first Frattini layer of the free kernel. Finite-rank kernels always satisfy the criterion, so a dimension drop requires an infinite-rank kernel, while the criterion remains stable under passage to open subgroups. On the fixed-point-free locus WD=0, the remaining dimension-one/two alternative is governed by H1(D,W∨) together with the Hochschild--Serre transgression. Dualizing the latter produces a relation-defect class in the topological coinvariants of W, which admits an explicit description in terms of a lifted defining relation of the Demuškin quotient. This yields a complete criterion for the lower-dimensional branch and separates the top-degree fixed-point obstruction from the extension data controlling dimensions one and two.
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BibTeX
@misc{palaisti2026detecting,
title = {{Detecting Cohomological Dimension Three in Free-by-Demuškin Pro-$p$ Groups}},
author = {Marina Palaisti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00021},
eprint = {2610.00021},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00021},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Detecting Cohomological Dimension Three in Free-by-Demuškin Pro-p Groups
AU - Marina Palaisti
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.00021
UR - https://arxiv.org/abs/2610.00021
AB - Let 1⟶ N⟶ G⟶ D⟶1 be exact, with N≠1 free pro-p of arbitrary rank and D Demuškin. Writing W=N/Φ(N), we prove H^3(G,𝔽_p)^∨≃ W^D, cd_p G=3 if and only if W^D≠0. Thus dimension three is detected on the first Frattini layer of the free kernel. Finite-rank kernels always satisfy the criterion, so a dimension drop requires an infinite-rank kernel, while the criterion remains stable under passage to open subgroups. On the fixed-point-free locus W^D=0, the remaining dimension-one/two alternative is governed by H^1(D,W^∨) together with the Hochschild--Serre transgression. Dualizing the latter produces a relation-defect class in the topological coinvariants of W, which admits an explicit description in terms of a lifted defining relation of the Demuškin quotient. This yields a complete criterion for the lower-dimensional branch and separates the top-degree fixed-point obstruction from the extension data controlling dimensions one and two.
ER -