{
  "abstract": "We show that von Neumann algebras of fundamental groups of closed orientable surfaces of genus $g\\geq2$ are free group factors on $2g-1$generators. The key technical ingredient involves a proof that the element $w=ABA^{-1}B^{-1}$ of the free group $\\mathbb{F}_{2}=\\langle A,B\\rangle$ is freely complemented in the group factor: $L(\\mathbb{F}_{2})=W^{*}(w)*W^{*}(v)$ for some Haar unitary $v\\in L(\\mathbb{F}_{2})$ that is freely independent from $w$. Combined with previous results, we conclude that for an arbitrary finitely generated torsion-free non-elementary discrete subgroup $Γ\\subset PSL_{2}(\\mathbb{R})$, $L(Γ)$ is a free group factor, settling a conjecture of de la Harpe and Voiculescu. This result was obtained using OpenAI's ChatGPT Pro 6.0.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11074",
  "author_total": 1,
  "authors": [
    {
      "name": "D. Shlyakhtenko"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d0s2zf1xnkw8m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11074",
  "fulltext_url": null,
  "id": "d0s2zf1xnkw8m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11074",
  "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.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7212255209",
  "pdf_url": "https://arxiv.org/pdf/2609.11074",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d0s2zf1xnkw8m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11074",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11074"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the II$_{1}$ Factors of Fuchsian Groups",
  "title_text": "On the II_1 Factors of Fuchsian Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/d0s2zf1xnkw8m",
  "venue": "arXiv",
  "versions": []
}