{
  "abstract": "Let $\\mathcal{N}$ be a non-empty class of nilpotent groups closed under taking subgroups, quotient groups, finite direct products, and unrestricted direct powers of the form $\\prod_{x \\in X}Y_{x}$, where $X \\in \\mathcal{N}$ and $Y_{x}$ ($x \\in X$) are isomorphic copies of a group $Y \\in \\mathcal{N}$. Let also the symbols $\\mathcal{N}^{\\mathfrak{tf}}$ and $\\mathit{Poly}_{k}\\mathcal{N}^{\\mathfrak{tf}}$ denote the class of all torsion-free $\\mathcal{N}$-groups and the class of polynilpotent groups such that $X \\in \\mathit{Poly}_{k}\\mathcal{N}^{\\mathfrak{tf}}$ if and only if $X$ has a subnormal series of length at most $k$ whose factors belong to $\\mathcal{N}^{\\mathfrak{tf}}$. We prove that if all free groups are residually $\\mathcal{N}^{\\mathfrak{tf}}$-groups and $2 \\leqslant n < \\infty$, then any tree product of $n$ residually $\\mathcal{N}^{\\mathfrak{tf}}$-groups with non-trivial torsion-free locally cyclic edge subgroups is residually a $(\\mathit{Poly}_{n+1}\\mathcal{N}^{\\mathfrak{tf}})$-group. In particular, all cyclically-pinched one-relator groups are residually finitely generated $(\\mathit{Poly}_{3}\\overline{\\mathcal{N}}\\mbox{}^{\\mathfrak{tf}})$-groups, where $\\overline{\\mathcal{N}}\\mbox{}^{\\mathfrak{tf}}$ is the class of all torsion-free nilpotent groups. We prove also that any tree product of $n$ $\\mathcal{N}$-groups with non-trivial torsion-free locally cyclic edge subgroups has a homomorphism onto a $(\\mathit{Poly}_{n-1}\\mathcal{N}^{\\mathfrak{tf}})$-group that is injective on all edge subgroups.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23126",
  "author_total": 1,
  "authors": [
    {
      "name": "E. V. Sokolov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a0zmv4j9tx9sn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23126",
  "fulltext_url": null,
  "id": "a0zmv4j9tx9sn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23126",
  "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",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7222916240",
  "pdf_url": "https://arxiv.org/pdf/2610.23126",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a0zmv4j9tx9sn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.23126",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23126"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the residual polynilpotence of tree products of groups with locally cyclic edge subgroups",
  "title_text": "On the residual polynilpotence of tree products of groups with locally cyclic edge subgroups",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/a0zmv4j9tx9sn",
  "venue": "arXiv",
  "versions": []
}