  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.
