Sharp Gromov Hyperbolicity Constants for Cartesian Products of Finitely Many Trees
Abstract
Let X=T1×⋯× Tk, k≥2, be a Cartesian product of finite nontrivial trees with unit edges, and write di=diam Ti, S=∑idi and M=maxidi. We prove that the thin-triangle hyperbolicity constant of the geodesic realization of X is Rips(X)=min{S/2, S−M+1/2}, and that the four-point hyperbolicity constant of its vertex metric is the value 𝒫(d1,…,dk) = maxE⊆{1,…,k} min{ ∑i∈ E di, ∑i∉ E di }. 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≤ 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≥3 the first formula exhibits a many-factor balancing phenomenon: several short factors can collectively balance the longest one and force the universal bound 1/2diam X to be attained. The proofs are self-contained and rest on an exact distance formula for the geodesic realization of X. For the ℓ1-product of the factors, i.e.\ the ℓ1-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 ℓ1-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.
Related
- Gromov hyperbolicity and uniformity
- Non-comparable invariant metrics and failure of Gromov hyperbolicity on a real-analytic pseudoconvex domain
- Optimal Linear Dependence on Boundary Type for Local Gromov Hyperbolicity of the Kobayashi Metric
- Stabilizing Metrics Induced by Logarithmic Oscillation in Metric Spaces and Gromov Hyperbolicity
- Gromov hyperbolicity, finite type, and subellipticity on ℂ-convex domains
BibTeX
@misc{michel2026sharp,
title = {{Sharp Gromov Hyperbolicity Constants for Cartesian Products of Finitely Many Trees}},
author = {Junior Michel and Anier Soria Lorente and Jean Marie-Vilaire},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24253},
eprint = {2610.24253},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24253},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp Gromov Hyperbolicity Constants for Cartesian Products of Finitely Many Trees
AU - Junior Michel
AU - Anier Soria Lorente
AU - Jean Marie-Vilaire
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.24253
UR - https://arxiv.org/abs/2610.24253
AB - Let X=T_1×⋯× T_k, k≥2, be a Cartesian product of finite nontrivial trees with unit edges, and write d_i=diam T_i, S=∑_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{S/2, S−M+1/2}, and that the four-point hyperbolicity constant of its vertex metric is the value 𝒫(d_1,…,d_k) = max_E⊆{1,…,k} min{ ∑_i∈ E d_i, ∑_i∉ E d_i }. 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≤ 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≥3 the first formula exhibits a many-factor balancing phenomenon: several short factors can collectively balance the longest one and force the universal bound 1/2diam X to be attained. The proofs are self-contained and rest on an exact distance formula for the geodesic realization of X. For the ℓ_1-product of the factors, i.e.\ the ℓ_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 ℓ_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.
ER -