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 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. We give a positive answer to this question for a large class of boundaries satisfying one crucial hypothesis, the {\it antipodal property}. This gives a class of compact spaces called {\it 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 {\it 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 {\it antipodal spaces} and {\it 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.