Compact quasi-metric spaces as hyperbolic boundaries
For any hyperbolic graph, its hyperbolic boundary equipped with the Gromov distance is known to be a compact quasi-metric space. In this talk, we consider the opposite problem and more. By formulating certain classes of hyperbolic graphs with good properties on geodesics, we show that any given compact quasi-metric space K can be identified with the hyperbolic boundary \partial X of some graph X in these classes, where the key notion \index map" acts as the bridge. The relationship between the doubling property of K and the bounded degree property of X will be discussed. This is a joint work with Ka-Sing Lau and Xiang-Yang Wang.