It is well known that there is a canonical cross-ratio on the boundary of a CAT(-1) space, which can be defined using visual metrics. This generalizes to the class of ""boundary continuous"" Gromov hyperbolic spaces, i.e. Gromov hyperbolic spaces where the Gromov inner product extends continuously to the boundary, with the visual metrics being replaced by visual quasi-metrics. We call such a space ""good"" if in addition it is proper and geodesically complete.

For a good space, the boundary equipped with a visual quasi-metric is a ""quasi-metric antipodal space"": a compact space with a continuous quasi-metric of diameter one such that every point has an "" antipode"", i.e. a point at distance one from the given point.

Given a quasi-metric antipodal space Z, we show that there is a hyperbolic filling of Z by a good space M(Z), such that the identification between Z and the boundary of M(Z) is ""Moebius"", i.e. preserves cross-ratios. Moreover, the space M(Z) is maximal amongst all such hyperbolic fillings of Z by good spaces: it has the universal property that any such hyperbolic filling embeds isometrically into M(Z). The space M(Z) is the unique good space with this property, and is called a ""maximal Gromov hyperbolic space"".

We show that the functor sending Z to M(Z) gives an equivalence of categories between quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces; two maximal Gromov hyperbolic spaces are isometric if and only if their boundaries are Moebius homeomorphic. This generalizes an equivalence, due to Beyrer and Schroeder, between ultrametric spaces and trees.