{
  "abstract": "For $n\\ge 3$ and $c\\in(0,1)$, let $\\mathcal{GT}(n,c)$ denote the class of geodesic metric trees of valence at most $n$ whose branch points are uniformly relatively separated with constant $c$. We prove that $\\mathcal{GT}(n,c)$ has no bi-Lipschitz universal element. More precisely, we construct a family $(T_a)_{a\\in[1/4,1/3]}\\subset\\mathcal{GT}(n,c)$ such that, for every $n_M\\geq 3, c_M\\in(0,1)$ and every $M\\in\\mathcal{GT}(n_M, c_M)$, there are at most countably many parameters $a$ for which $T_a$ admits a bi-Lipschitz embedding into $M$, whereas each $T_a$ admits a bi-Lipschitz embedding into $\\mathbb R^2$. Thus the obstruction is neither dimensional nor caused by a failure of planar embeddability. This gives a negative answer to a question of Chrontsios-Garitsis, Ioannidis, and Vellis~\\cite[Question~1.11]{CGIV2024}. Furthermore, we show a complementary positive result for ultrametric spaces: every bounded ultrametric space $X$ admits a bi-Lipschitz embedding into every complete metric space $Y$ satisfying $\\dim_A X<\\dim_{LA} Y$ where $\\dim_A X$ and $\\dim_{LA} X$ are Assouad and lower Assouad dimensions, respectively.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09110",
  "author_total": 2,
  "authors": [
    {
      "name": "Sylvester Eriksson-Bique"
    },
    {
      "name": "Manisha Garg"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2vdnx0wgwh20q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09110",
  "fulltext_url": null,
  "id": "2vdnx0wgwh20q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09110",
  "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.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7212057981",
  "pdf_url": "https://arxiv.org/pdf/2609.09110",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2vdnx0wgwh20q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2609.09110",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09110"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On universal elements for doubling geodesic trees",
  "title_text": "On universal elements for doubling geodesic trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/2vdnx0wgwh20q",
  "venue": "arXiv",
  "versions": []
}