{
  "abstract": "A recent preprint constructs a finitely presented torsion-free group $G$ with a finite two-dimensional classifying space such that $\\mathbb F_2[G]$ contains nonzero zero divisors. We extend this to every odd prime $p$: there is a finitely presented torsion-free group $G_p$, again with a finite two-dimensional classifying space, such that $\\mathbb F_p[G_p]$ contains nonzero elements $α,β$ with $αβ=0$. Together with the case $p=2$, Kaplansky's zero-divisor conjecture therefore fails over every field of positive characteristic. The new ingredients are a signed version of the parity cancellation, made possible by a bipartite random model on one side; a type design built from the projective plane over $\\mathbb F_{p^3}$ with $p$ additional letters, which makes every cross-intersection congruent to $1$ modulo $p$; and a squared-word decay estimate for this design, proved with an explicit test vector. The geometric part of the argument is taken from the characteristic-two construction. The characteristic-zero case remains open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17662",
  "author_total": 1,
  "authors": [
    {
      "name": "S. V. Ivanov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/41n41rqw9ta7f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17662",
  "fulltext_url": null,
  "id": "41n41rqw9ta7f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17662",
  "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"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7221790634",
  "pdf_url": "https://arxiv.org/pdf/2610.17662",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/41n41rqw9ta7f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17662",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17662"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Zero divisors in group algebras of torsion-free groups in odd characteristic",
  "title_text": "Zero divisors in group algebras of torsion-free groups in odd characteristic",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/41n41rqw9ta7f",
  "venue": "arXiv",
  "versions": []
}