Polyhedral structure of maximal Gromov hyperbolic spaces with finite boundary
Kingshook Biswas, Arkajit Pal Choudhury
TL;DR
The paper solves the problem of describing maximal Gromov hyperbolic spaces $X$ with finite boundary by showing that, for $|\,\partial X\,|=n$, $X$ is isometric to a polyhedral complex $\mathcal{P}\subset \mathbb{R}^n$ with the $\ell_\infty$ metric, built by attaching $n$ half-lines to a compact core and whose combinatorics are governed by antipodal relations on $\partial X$. The boundary Möbius structure, encoded by cross-ratios, fully determines $X$ up to isometry in this maximal setting, and the spaces $\mathcal{M}(Z,\rho)$ arise as the upper bounds of all hyperbolic fillings of a finite antipodal boundary $Z$. The authors further show that maximal spaces are injective, relate these spaces to injective hulls of finite spheres, and develop a Teichmüller-type deformation theory $\mathcal{T}(Z)$ parameterized by antipodal data, including an explicit treatment of the 4-point boundary case. The results reveal that the geometry at infinity is trivial for finite boundary maximal spaces and provide a concrete, polyhedral, and combinatorial framework for understanding these spaces and their deformations, with several open problems guiding future work on polyhedral realizations, countable boundaries, and higher-point Teichmüller structures.
Abstract
The boundary $\partial X$ of a boundary continuous Gromov hyperbolic space $X$ carries a natural Moebius structure on the boundary. For a proper, geodesically complete, boundary continuous Gromov hyperbolic space $X$, the boundary $\partial X$ equipped with its cross-ratio is a particular kind of quasi-metric space, called a quasi-metric antipodal space. Given a quasi-metric antipodal space $Z$, one may consider the family of all hyperbolic fillings of $Z$. In \cite{biswas2024quasi} it was shown that this family has a unique upper bound $\mathcal{M}(Z)$ (with respect to a natural partial order on hyperbolic fillings of $Z$), which can be described explicitly in terms of the cross-ratio on $Z$. As shown in \cite{biswas2024quasi}, the spaces $\mathcal{M}(Z)$ constitute a natural class of spaces called maximal Gromov hyperbolic spaces. A natural problem is to describe explicitly the maximal Gromov hyperbolic spaces $X$ whose boundary $\partial X$ is finite. We show that for a maximal Gromov hyperbolic space $X$ with boundary $\partial X$ of cardinality $n$, the space $X$ is isometric to a finite polyhedral complex embedded in $(\mathbb{R}^n, ||\cdot||_{\infty})$ with cells of dimension at most $n/2$, given by attaching $n$ half-lines to vertices of a compact polyhedral complex. In particular the geometry at infinity of $X$ is trivial. The combinatorics of the polyhedral complex is determined by certain relations $R \subset \partial X \times \partial X$ on the boundary $\partial X$, called antipodal relations. In \cite{biswas2024quasi} it was shown that maximal Gromov hyperbolic spaces are injective metric spaces. We give a shorter, simpler proof of this fact in the case of spaces with finite boundary. We also consider the space of deformations of a maximal Gromov hyperbolic space with finite boundary, and define an associated Teichmuller space.