Quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces

Abstract

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 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 $\mathcal{M}(Z)$