Lazard's Realization Problem for N-Series: Bar Obstructions and Finite-Quotient Descent
Abstract
Lazard asked which N-series of a group arise from multiplicative filtrations of the integral group ring. Given an N-series H• of a group G, let A• be the filtration of the augmentation ideal spanned by products of elements x−1, x∈ Ha, weighted by a. Every realizing filtration contains A•, so H• is realizable exactly when the maps Hn/Hn+1→ An/An+1, h↦ h−1, are injective. We show that each kernel is the cokernel of an explicit map of normalized bar groups; for a finite group this makes realizability decidable by integer linear algebra. For a profinite group with a cofinal tower of finite quotients and an N-series induced from it, suppose h∈ Hn and h−1∈ An+1 in every finite quotient. Then h−1∈ An+1 in the discrete group ring exactly when the widths of its finite-level Losey expressions are bounded. This descent can fail. Using Tahara's class-three $2$-groups and the Hartl-Mikhailov-Passi description of the fourth dimension subgroup, we construct a countable product P of finite $2$-groups. It contains an element of γ3(P) that lies in the fourth dimension subgroup of every finite subproduct of P but not in D4(P).
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BibTeX
@misc{ma2026lazards,
title = {{Lazard's Realization Problem for N-Series: Bar Obstructions and Finite-Quotient Descent}},
author = {Chao Ma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.38002},
eprint = {2609.38002},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38002},
}
RIS
TY - UNPB
TI - Lazard's Realization Problem for N-Series: Bar Obstructions and Finite-Quotient Descent
AU - Chao Ma
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2609.38002
UR - https://arxiv.org/abs/2609.38002
AB - Lazard asked which N-series of a group arise from multiplicative filtrations of the integral group ring. Given an N-series H_• of a group G, let A_• be the filtration of the augmentation ideal spanned by products of elements x−1, x∈ H_a, weighted by a. Every realizing filtration contains A_•, so H_• is realizable exactly when the maps H_n/H_n+1→ A_n/A_n+1, h↦ h−1, are injective. We show that each kernel is the cokernel of an explicit map of normalized bar groups; for a finite group this makes realizability decidable by integer linear algebra. For a profinite group with a cofinal tower of finite quotients and an N-series induced from it, suppose h∈ H_n and h−1∈ A_n+1 in every finite quotient. Then h−1∈ A_n+1 in the discrete group ring exactly when the widths of its finite-level Losey expressions are bounded. This descent can fail. Using Tahara's class-three $2$-groups and the Hartl-Mikhailov-Passi description of the fourth dimension subgroup, we construct a countable product P of finite $2$-groups. It contains an element of γ_3(P) that lies in the fourth dimension subgroup of every finite subproduct of P but not in D_4(P).
ER -