A Poincaré ball model for taxicab hyperbolic geometry

TL;DR

This work introduces the taxicab Poincaré ball $B_T^n$, equipping it with a tangent-norm $ig\\|v\\\|_{B_T} = \\frac{\\|v\\|_T}{1-\\|x\\|_T^2}$ to define lengths of absolutely continuous curves. Central to the geometry are the minimal points $m(p,q)$ and the L-shaped geodesics $\\lambda_{p,q}$, which yield the distance formula $d(p,q) = \\tanh^{-1}(\\|p\\|_T) + \\tanh^{-1}(\\|q\\|_T) - 2\\tanh^{-1}(\\|m\\|_T)$, and a precise description of isometries via the hyperoctahedral group $H_n$. The space is geodesic but not homogeneous or median; nevertheless, it is Gromov-hyperbolic with optimal constant $\\ln(3)$, and it forms a coarse median space. Overall, the paper establishes a concrete taxicab hyperbolic model, analyzes its geometric structure (distance, spheres, intervals, and symmetries), and outlines future directions for alternative models and extensions.

Abstract

Taxicab space is a modification of Euclidean space that uses an alternative notion of distance. Similarly, the Poincaré ball is a model of hyperbolic geometry that consists of a subset of Euclidean space with an alternative notion of distance. In this paper, we merge these two variations to create a taxicab version of the Poincaré ball. We determine the isometry group for this new space and show that this space is hyperbolic in the sense of Gromov.

Figures (8)

• Figure 1: Some points in $B_T^2$, the minimal points associated to various pairs of points, and the corresponding L-shaped curves.
• Figure 2: Absolutely continuous functions $f$ in gray, their cumulative minimum functions $\underline{f}$ in solid black, and their residual minimum functions $\overline{f}$ in dashed black. In the top image, $f$ is positive. In the bottom image, $f$ changes sign.
• Figure 3: Various examples when $n = 2$ of curves $\gamma$ in gray, from $p$ to $q$, and the adjustments $\underline{\gamma}$, solid black and $\overline{\gamma}$, dashed black, which are concatenated to produce $\widetilde{\gamma}$. In (a), $p$ and $q$ lie in the same quadrant with $p$ lying beyond $q$. In this example, $\overline{\gamma}$ is stationary so $\widetilde{\gamma} \sim \underline{\gamma}$. In (b), $p$ and $q$ lie in the same quadrant and neither point is beyond the other. In (c) $p$ and $q$ lie in adjacent quadrants and it happens that $m(p, q) = m(\gamma)$ which need not always occur. In $(d)$, $p$ and $q$ lie in opposite quadrants. Again, $m(p, q) = m(\gamma)$ which turns out always to be true when $p$ and $q$ lie in opposite orthants.
• Figure 4: An example when $n = 2$ showing the sequence of steps adjusting the curve $\gamma$ through $\widetilde{\gamma}$ and $\varphi$ to $\lambda_{p, q}$. In each step, the length of the new curve is no greater than that of the previous curve. In the third step, the portions of $\varphi$ from $m(p, q)$ to $m(\gamma)$ and back are separated slightly for visual effect.
• Figure 5: Various intervals in $B_T^2$.

Theorems & Definitions (23)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Proposition 2.1
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['minimizerthm']}
• proof
• Lemma 4.1