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.

度量空间的双曲填充是一种众所周知的工具，用于证明将格罗莫夫双曲空间边界之间的拟莫比乌斯映射扩展到空间之间的拟等距的结果。对于格罗莫夫双曲空间X的边界的双曲填充Y ，边界\(\partial Y\)和\(\partial X\)之间具有拟莫比乌斯辨识。对于 CAT(-1) 空间，以及更一般的边界连续格罗莫夫双曲空间，可以将边界上的拟莫比斯结构细化为莫比斯结构。那么很自然地就会问是否存在边界连续格罗莫夫双曲空间对边界的函子双曲填充，并且边界之间的识别不仅是准莫比乌斯，而且实际上是莫比乌斯。填充应该是函数式的，因为边界之间的莫比乌斯同胚应该引起填充之间的等距。对于满足一个关键假设（对映性质）的一大类边界，我们对这个问题给出了肯定的答案。这给出了一类称为拟度量对映空间的紧空间。对于任何这样的空间Z，我们给出边界连续格罗莫夫双曲空间\(\mathcal {M}(Z)\)的函子构造以及其与Z的边界的莫比乌斯辨识。空间\(\mathcal {M}(Z)\)在Z的所有填充中是最大的。这些空间\(\mathcal {M}(Z)\)实际上给出了称为最大格罗莫夫双曲空间的自然类空间的所有示例。我们证明了拟度量对映空间和最大格罗莫夫双曲空间之间的范畴等价。这是我们在称为对映空间和最大格罗莫夫乘积空间的某些空间的更大类别之间证明的更一般等价的一部分。我们证明格罗莫夫乘积空间X的单射包与最大格罗莫夫乘积空间\(\mathcal {M}(Z)\)等距，其中Z是X的边界。我们还表明，格罗莫夫积空间是单射的当且仅当它是最大的。