{
  "abstract": "We exhibit two finitely generated residually finite groups $G_1$ and $G_2$ with isomorphic profinite completions $\\widehat{G_1} \\cong \\widehat{G_2}$, such that $G_1$ has a faithful representation into $GL(d,\\mathbb{C})$ but $G_2$ is not linear. This answers a question of Andrei Jaikin-Zapirain, Alexander Lubotzky, and Martin Bridson negatively. We also present two finitely generated residually finite groups $H_1$ and $H_2$ with isomorphic profinite completions $\\widehat{H_1} \\cong \\widehat{H_2}$ satisfying the following. For every field $k$ with characteristic $p \\geq 5$ and transcendence degree $r \\geq 2$, there exists $d>0$ such that $H_1$ has a faithful representation into $GL(d,k)$. However, $H_2$ admits no faithful finite-dimensional representation over any field of positive characteristic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01672",
  "author_total": 1,
  "authors": [
    {
      "name": "Wonyong Jang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5r7m0w5tn54ma.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01672",
  "fulltext_url": null,
  "id": "5r7m0w5tn54ma",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01672",
  "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.GT"
    ],
    [
      "arxiv",
      "math.RT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01672",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5r7m0w5tn54ma.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01672",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01672"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A faithful representation and profinite completion",
  "title_text": "A faithful representation and profinite completion",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/5r7m0w5tn54ma",
  "venue": "arXiv",
  "versions": []
}