{
  "abstract": "Let $X=T_1\\times\\cdots\\times T_k$, $k\\ge2$, be a Cartesian product of finite nontrivial trees with unit edges, and write $d_i=\\diam T_i$, $S=\\sum_id_i$ and $M=\\max_id_i$. We prove that the thin-triangle hyperbolicity constant of the geodesic realization of $X$ is \\[ \\Rips(X)=\\min\\Bigl\\{\\frac S2,\\ S-M+\\frac12\\Bigr\\}, \\] and that the four-point hyperbolicity constant of its vertex metric is the value \\[ \\mathcal{P}(d_1,\\dots,d_k) = \\max_{E\\subseteq\\{1,\\dots,k\\}} \\min\\left\\{ \\sum_{i\\in E} d_i,\\, \\sum_{i\\notin E} d_i \\right\\}. \\] of the best two-way partition of the diameters. Both constants therefore depend only on the diameter vector, and their difference is governed by the partition imbalance $η$ of that vector: it equals $η/2$ when $M\\le S-M$ and $1/2$ otherwise. In particular the two constants coincide if and only if the diameters admit a perfect partition, and their difference is unbounded. For $k\\ge3$ the first formula exhibits a many-factor balancing phenomenon: several short factors can collectively balance the longest one and force the universal bound $\\frac12\\diam X$ to be attained. The proofs are self-contained and rest on an exact distance formula for the geodesic realization of $X$. For the $\\ell_1$-product of the factors, i.e.\\ the $\\ell_1$-metric on the associated cube complex, we obtain $\\Rips=\\min\\{S/2,S-M\\}$, which identifies the half-unit above as a pure $1$-skeleton effect, and we show that the four-point formula holds for $\\ell_1$-products of arbitrary $0$-hyperbolic metric spaces, in particular for weighted trees and $\\R$-trees. We also treat products with unbounded factors and record algorithmic consequences.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24253",
  "author_total": 3,
  "authors": [
    {
      "name": "Junior Michel"
    },
    {
      "name": "Anier Soria Lorente"
    },
    {
      "name": "Jean Marie-Vilaire"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/13vngx4tnjfhz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24253",
  "fulltext_url": null,
  "id": "13vngx4tnjfhz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24253",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7223037091",
  "pdf_url": "https://arxiv.org/pdf/2610.24253",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/13vngx4tnjfhz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24253",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24253"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Sharp Gromov Hyperbolicity Constants for Cartesian Products of Finitely Many Trees",
  "title_text": "Sharp Gromov Hyperbolicity Constants for Cartesian Products of Finitely Many Trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/13vngx4tnjfhz",
  "venue": "arXiv",
  "versions": []
}