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On universal elements for doubling geodesic trees

  • Sylvester Eriksson-Bique
  • Manisha Garg
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Abstract

For n≥ 3 and c∈(0,1), let 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 𝒢𝒯(n,c) has no bi-Lipschitz universal element. More precisely, we construct a family (Ta)a∈[1/4,1/3]⊂𝒢𝒯(n,c) such that, for every nM≥ 3, cM∈(0,1) and every M∈𝒢𝒯(nM, cM), there are at most countably many parameters a for which Ta admits a bi-Lipschitz embedding into M, whereas each Ta admits a bi-Lipschitz embedding into ℝ2. 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 dimA X<dimLA Y where dimA X and dimLA X are Assouad and lower Assouad dimensions, respectively.

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Cite

BibTeXRIS
BibTeX
@misc{erikssonbique2026universal,
  title = {{On universal elements for doubling geodesic trees}},
  author = {Sylvester Eriksson-Bique and Manisha Garg},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09110},
  eprint = {2609.09110},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09110},
}
RIS
TY  - UNPB
TI  - On universal elements for doubling geodesic trees
AU  - Sylvester Eriksson-Bique
AU  - Manisha Garg
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.09110
UR  - https://arxiv.org/abs/2609.09110
AB  - For n≥ 3 and c∈(0,1), let 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 𝒢𝒯(n,c) has no bi-Lipschitz universal element. More precisely, we construct a family (T_a)_a∈[1/4,1/3]⊂𝒢𝒯(n,c) such that, for every n_M≥ 3, c_M∈(0,1) and every M∈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 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.
ER  -