{
  "abstract": "The $n$-$(n+1)$-$(n+2)$ theorem, recently established by Cohen and Shusterman, says that if two groups of type $\\mathrm{F}_{n+1}$ map onto a common quotient $Q$ of type $\\mathrm{F}_{n+2}$ and one of the two kernels is of type $\\mathrm{F}_n$, then the associated fibre product is of type $\\mathrm{F}_{n+1}$. We prove a stronger, symmetric variant of this theorem when $Q$ is virtually nilpotent. More precisely, for $i\\in\\{1,2\\}$, let $1\\to N_i\\to Γ_i\\overset{\\smallπ_i}{\\to} Q\\to 1$ be short exact sequences of groups and let $P=\\{(γ_1,γ_2)\\inΓ_1\\timesΓ_2\\colon π_1(γ_1)=π_2(γ_2)\\}$ be their fibre product. Let $k,l,m,n\\in\\mathbb N_0$, assume that $N_1$ and $N_2$ are of type $\\mathrm{FP}_k$ and $\\mathrm{FP}_l$, respectively, and that $Γ_1$ and $Γ_2$ are of type $\\mathrm{F}_m$ and $\\mathrm{F}_n$, respectively. We prove that $P$ is of type $\\mathrm{F}_{\\min\\{k+l+1,m,n\\}}$. The same formula holds with the homological finiteness properties $\\mathrm{FP}_r$ in place of $\\mathrm{F}_r$. This provides an alternative proof of the homotopical and homological Virtual Surjections Theorem, which gives a finiteness criterion for subgroups of direct products in terms of their embedding in the ambient product.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08911",
  "author_total": 1,
  "authors": [
    {
      "name": "Benno Kuckuck"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/08qcmdn7rynh3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08911",
  "fulltext_url": null,
  "id": "08qcmdn7rynh3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08911",
  "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": "W7212028454",
  "pdf_url": "https://arxiv.org/pdf/2609.08911",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/08qcmdn7rynh3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2609.08911",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08911"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Finiteness Properties of Fibre Products over Virtually Nilpotent Quotients",
  "title_text": "Finiteness Properties of Fibre Products over Virtually Nilpotent Quotients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/08qcmdn7rynh3",
  "venue": "arXiv",
  "versions": []
}