Sharp Gromov Hyperbolicity Constants for Cartesian Products of Finitely Many Trees

Abstract

Let $X=T_1\times \cdots \times T_k$, $k\ge 2$, be a Cartesian product of finite nontrivial trees with unit edges, and write $d_i=\diam T_i$, $S={\sum }_i{d}_i$ and $M={\max }_i{d}_i$. 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 $\eta$ of that vector: it equals $\eta /2$ 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 $k\ge 3$ 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 $\mathbb{R}$-trees. We also treat products with unbounded factors and record algorithmic consequences.