Quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces

Hyperbolic fillings of metric spaces are a well-known tool for proving results on extending quasi-Moebius maps between boundaries of Gromov hyperbolic spaces to quasi-isometries between the spaces. For a hyperbolic filling Y of the boundary of a Gromov hyperbolic space X, one has a quasi-Moebius identification between the boundaries \(\partial Y\) and \(\partial X\). For CAT(-1) spaces, and more generally boundary continuous Gromov hyperbolic spaces, one can refine the quasi-Moebius structure on the boundary to a Moebius structure. It is then natural to ask whether there exists a functorial hyperbolic filling of the boundary by a boundary continuous Gromov hyperbolic space with an identification between boundaries which is not just quasi-Moebius, but in fact Moebius. The filling should be functorial in the sense that a Moebius homeomorphism between boundaries should induce an isometry between there fillings. We give a positive answer to this question for a large class of boundaries satisfying one crucial hypothesis, the antipodal property. This gives a class of compact spaces called quasi-metric antipodal spaces. For any such space Z, we give a functorial construction of a boundary continuous Gromov hyperbolic space \(\mathcal {M}(Z)\) together with a Moebius identification of its boundary with Z. The space \(\mathcal {M}(Z)\) is maximal amongst all fillings of Z. These spaces \(\mathcal {M}(Z)\) give in fact all examples of a natural class of spaces called maximal Gromov hyperbolic spaces. We prove an equivalence of categories between quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces. This is part of a more general equivalence we prove between the larger categories of certain spaces called antipodal spaces and maximal Gromov product spaces. We prove that the injective hull of a Gromov product space X is isometric to the maximal Gromov product space \(\mathcal {M}(Z)\), where Z is the boundary of X. We also show that a Gromov product space is injective if and only if it is maximal.