Canonical Kummerian Kernels in 1-Smooth Pro-p Poincaré Duality Groups of Dimension Three
Abstract
Let (G,θ) be a finitely generated 1-smooth oriented pro-p group with torsion-free orientation, and suppose that G is a Poincaré duality pro-p group of dimension three. The orientation determines a canonical closed normal subgroup Kθ(G) contained in the Frattini subgroup, and for a Kummerian pair the quotient G/Kθ(G) is the maximal θ-abelian quotient. We prove that Kθ(G) finitely generated Longrightarrow Kθ(G)=1. Thus every non-θ-abelian 1-smooth PD3 pro-p group has an infinitely generated canonical kernel, where ``infinitely generated'' means not topologically finitely generated. The proof combines the normal-subgroup theorem for pro-p PD3 groups with the classification of p-adic analytic 1-smooth pro-p groups. We also obtain a cohomological description of the canonical extension from the Hochschild--Serre five-term sequence and Poincaré duality. Writing K=Kθ(G), Q=G/K, and V=H1(G,𝔽p), we identify H1(K,𝔽p)Q with the kernel of the map bigwedge2V→ V* induced by the triple cup-product form of G. Consequently, whenever K≠1, its first cohomology is infinite-dimensional, whereas its Q-invariant subspace is nonzero and finite-dimensional. We obtain exact dimension restrictions in small generator ranks and show that a non-θ-abelian group with 2≤ d(G)≤4 is not Bloch--Kato. In the finitely generated-kernel case, G is a three-dimensional uniform θ-abelian group with exterior mod-p cohomology.
Related
- Poincaré duality for k-A Lie superalgebras
- Uniqueness of differential Poincaré duality models
- Poincaré duality for pro-étale Qp-local systems
- Poincaré Duality and Quadratic Refinements over Laurent Rings
- Poincaré's Inequality on Sliding-Window Occupation Count Kernels and the Optimal Normalized Spectral-Gap Constant
BibTeX
@misc{palaisti2026canonical,
title = {{Canonical Kummerian Kernels in 1-Smooth Pro-$p$ Poincaré Duality Groups of Dimension Three}},
author = {Marina Palaisti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20915},
eprint = {2610.20915},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20915},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Canonical Kummerian Kernels in 1-Smooth Pro-p Poincaré Duality Groups of Dimension Three
AU - Marina Palaisti
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20915
UR - https://arxiv.org/abs/2610.20915
AB - Let (G,θ) be a finitely generated 1-smooth oriented pro-p group with torsion-free orientation, and suppose that G is a Poincaré duality pro-p group of dimension three. The orientation determines a canonical closed normal subgroup K_θ(G) contained in the Frattini subgroup, and for a Kummerian pair the quotient G/K_θ(G) is the maximal θ-abelian quotient. We prove that K_θ(G) finitely generated Longrightarrow K_θ(G)=1. Thus every non-θ-abelian 1-smooth PD^3 pro-p group has an infinitely generated canonical kernel, where ``infinitely generated'' means not topologically finitely generated. The proof combines the normal-subgroup theorem for pro-p PD^3 groups with the classification of p-adic analytic 1-smooth pro-p groups. We also obtain a cohomological description of the canonical extension from the Hochschild--Serre five-term sequence and Poincaré duality. Writing K=K_θ(G), Q=G/K, and V=H^1(G,𝔽_p), we identify H^1(K,𝔽_p)^Q with the kernel of the map bigwedge^2V→ V^* induced by the triple cup-product form of G. Consequently, whenever K≠1, its first cohomology is infinite-dimensional, whereas its Q-invariant subspace is nonzero and finite-dimensional. We obtain exact dimension restrictions in small generator ranks and show that a non-θ-abelian group with 2≤ d(G)≤4 is not Bloch--Kato. In the finitely generated-kernel case, G is a three-dimensional uniform θ-abelian group with exterior mod-p cohomology.
ER -