{
  "abstract": "A classical uniformization theorem of Ahlfors from 1963 asserts that every bounded turning planar arc is the image of the unit segment under a quasiconformal self-map of the plane. We prove an analogous uniformization theorem for planar trees by showing that every planar quasiconformal tree with uniform branch separation is quasiconformally equivalent to a quasiconvex planar tree with uniform branch separation and Hausdorff dimension arbitrarily close to 1. Along the way, we show that every quasisymmetric (resp. bi-Lipschitz) homeomorphism between planar quasiconformal trees that respects the counterclockwise orientation of the branches at every branch point can be extended quasisymmetrically (resp. bi-Lipschitz) to the whole plane. As a corollary of the extension theorem we construct a large class of quadrivalent quasiconformal trees, which includes the Vicsek fractal, so that each tree in this class is the Julia set of a uniformly quasiregular map of $\\mathbb{R}^2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14236",
  "author_total": 2,
  "authors": [
    {
      "name": "Fotis Ioannidis"
    },
    {
      "name": "Vyron Vellis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a9kba75yk3wsd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14236",
  "fulltext_url": null,
  "id": "a9kba75yk3wsd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14236",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7221814254",
  "pdf_url": "https://arxiv.org/pdf/2610.14236",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a9kba75yk3wsd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.14236",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14236"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Quasiconformal uniformization and extension for planar trees",
  "title_text": "Quasiconformal uniformization and extension for planar trees",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/a9kba75yk3wsd",
  "venue": "arXiv",
  "versions": []
}