The boundary \(\partial X\) of a Gromov hyperbolic space X carries a natural quasi-Moebius structure, induced by the family of visual quasi-metrics on the boundary \(\rho _x = e^{-(\cdot |\cdot )_x}, x \in X\) , which are all quasi-Moebius equivalent to each other. For boundary continuous Gromov hyperbolic spaces the visual quasi-metrics are Moebius equivalent to each other and one has a finer Moebius structure on the boundary. In this context, a natural problem is to try to reconstruct the space X from its boundary \(\partial X\) . 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 Biswas (Geom. Dedicata 218(2):53, 2024) 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. A proper, geodesically complete, boundary continuous Gromov hyperbolic X is called a maximal Gromov hyperbolic space if it is an upper bound for the family of fillings of its boundary \(\partial X\) . In Biswas (Geom. Dedicata 218(2):53, 2024), it was shown that the maximal Gromov hyperbolic spaces are precisely those of the form \({\mathcal {M}}(Z)\) for some quasi-metric antipodal space Z. 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; each cell of the complex corresponds to an antipodal relation. In Biswas (Geom. Dedicata 218(2):53, 2024) 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, using the fact that their geometry at infinity is trivial. We also consider the space of deformations of a maximal Gromov hyperbolic space with finite boundary, and define an associated Teichmuller space and mapping class group.
