{
  "abstract": "Ghosh, Meiri and Sarnak showed that the commutator equation over $\\mathrm{SL}_2(\\mathbb{Z})$ satisfies a profinite local to global principle, and that for a prime $\\ell\\equiv\\pm1$ mod $5$ it fails for $\\mathrm{SL}_2(\\mathbb{Z}[1/\\ell])$ with a reciprocity obstruction. We extend the failure to surface groups. For the genus-$2$ surface group $Γ_2$ we construct an explicit representation $ρ:Γ_2\\to\\mathrm{SL}_2(\\mathcal{O}_{K,S})$, where $K=\\mathbb{Q}(\\sqrt{5})$ and $\\mathcal{O}_{K,S}=\\mathbb{Z}[\\frac{1+\\sqrt{5}}{2},\\frac{1}{10}]$, whose quotients modulo the maximal ideals are the full groups $\\mathrm{SL}_2(\\mathbb{F}_q)$ except in residue characteristic $3$, where the quotient has order $24$. For every $k\\ge2$ the word $[a_1,b_1]^k$ is not a commutator in $Γ_2$, while its image is a commutator in every one of these quotients. The same holds for every surface group $Γ_g$ with $g\\ge2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36497",
  "author_total": 1,
  "authors": [
    {
      "name": "Jianchun Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/343mxmagx9hc9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36497",
  "fulltext_url": null,
  "id": "343mxmagx9hc9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36497",
  "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.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7224124930",
  "pdf_url": "https://arxiv.org/pdf/2610.36497",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/343mxmagx9hc9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36497",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36497"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Failure of the congruence Hasse principle for commutators in orientable surface groups",
  "title_text": "Failure of the congruence Hasse principle for commutators in orientable surface groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/343mxmagx9hc9",
  "venue": "arXiv",
  "versions": []
}