Quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces

TL;DR

This paper constructs a functorial hyperbolic filling for boundary continuous Gromov hyperbolic spaces with antipodal properties, establishing an equivalence between certain boundary spaces and maximal hyperbolic spaces, and characterizing injective Gromov product spaces.

Contribution

It introduces a functorial construction of hyperbolic fillings for antipodal boundary spaces and proves an equivalence of categories with maximal Gromov hyperbolic spaces, including characterizations of injective spaces.

Findings

Constructs maximal Gromov hyperbolic spaces from antipodal boundary spaces.

Establishes an equivalence of categories between these boundary spaces and hyperbolic spaces.

Shows that injective Gromov product spaces are exactly the maximal ones.

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 {\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.