{
  "abstract": "Do Kaplansky's group-ring conjectures hold for torsion-free word-hyperbolic groups? We show that all four fail in positive characteristic. For every prime $p$ there is a finitely presented, torsion-free, word-hyperbolic group $G$ with a finite two-dimensional classifying space and $a,b\\in\\mathbb{F}_p[G]$ with $ab=1\\neq ba$. So, over every field of characteristic $p$, the group algebra of $G$ is not directly finite and has a non-trivial idempotent, zero divisor and unit, and $G$ is neither sofic nor surjunctive. For $p=2$, $G$, $a$ and $b$ come from OpenAI's random construction, and we add that $G$ is hyperbolic with probability $1-o(1)$, as are OpenAI's zero-divisor groups. For odd $p$ the groups are new: they adapt OpenAI's direct-finiteness construction with a signed cancellation, found first by S. V. Ivanov and independently by us, via an exact solution of a colour-balance problem and a Perron certificate. As far as we know, they are the first torsion-free counterexamples to direct finiteness and to the idempotent conjecture in odd characteristic. Hyperbolicity comes from one deterministic criterion: if a labelled graph is pattern-free, the group obtained by coning off its components is aspherical and has a linear isoperimetric inequality. We prove it by running OpenAI's exclusion of reduced spheres on van Kampen discs, an accounting OpenAI used for another complex. For every prime $p$, the existence of $G$, $a$ and $b$ and the failure of all four conjectures are formalized in Lean 4, including existence forms of the random-graph estimates used. In characteristic zero, where the idempotent and direct-finiteness conjectures hold for torsion-free hyperbolic groups, we show that no cancellation built into the vertex types survives the decay estimate that makes the groups torsion-free. OpenAI's characteristic-two counterexample with torsion is a CAT(0) group.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04599",
  "author_total": 2,
  "authors": [
    {
      "name": "Nidhal Mghirbi"
    },
    {
      "name": "Seth Douglas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/17d7wmgeact7j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04599",
  "fulltext_url": null,
  "id": "17d7wmgeact7j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04599",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.RA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.04599",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/17d7wmgeact7j.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.04599",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04599"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Kaplansky's conjectures fail for torsion-free hyperbolic groups in positive characteristic",
  "title_text": "Kaplansky's conjectures fail for torsion-free hyperbolic groups in positive characteristic",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/17d7wmgeact7j",
  "venue": "arXiv",
  "versions": []
}