View article

The aims of this paper are to establish connections between a metric space X and the large-scale geometry (in the sense of Gromov) of the hyperspace E (X) of its nondegenerate closed bounded subsets and to study mappings on X in terms of the induced mappings on E (X). The metric space X can be identified with the boundary of E (X) when the latter is equipped with the Hausdorff metric, but stronger relationships between X and E (X) are obtained when the hyperspace E (X) is hyperbolized and the space X is identified with its boundary at infinity: a priori weak conditions on E (X) are strengthened at the boundary at infinity. The basic tool for studying such relationships is Gromov’s theory of negatively curved spaces [22]. These spaces, known as Gromov hyperbolic spaces, are important in many areas of analysis and geometry, including geometric function theory, geometric group theory, and analysis on metric …