{
  "abstract": "Lazard asked which $N$-series of a group arise from multiplicative filtrations of the integral group ring. Given an $N$-series $H_\\bullet$ of a group $G$, let $A_\\bullet$ be the filtration of the augmentation ideal spanned by products of elements $x-1$, $x\\in H_a$, weighted by $a$. Every realizing filtration contains $A_\\bullet$, so $H_\\bullet$ is realizable exactly when the maps $H_n/H_{n+1}\\to A_n/A_{n+1}$, $h\\mapsto h-1$, 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 $h\\in H_n$ and $h-1\\in A_{n+1}$ in every finite quotient. Then $h-1\\in A_{n+1}$ 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 $D_4(P)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38002",
  "author_total": 1,
  "authors": [
    {
      "name": "Chao Ma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f4xyemphyzytq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38002",
  "fulltext_url": null,
  "id": "f4xyemphyzytq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38002",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7214899591",
  "pdf_url": "https://arxiv.org/pdf/2609.38002",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f4xyemphyzytq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2609.38002",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38002"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Lazard's Realization Problem for N-Series: Bar Obstructions and Finite-Quotient Descent",
  "title_text": "Lazard's Realization Problem for N-Series: Bar Obstructions and Finite-Quotient Descent",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/f4xyemphyzytq",
  "venue": "arXiv",
  "versions": []
}