Embeddings of Gromov Hyperbolic Spaces

Abstract

It is shown that a Gromov hyperbolic geodesic metric space X with bounded growth at some scale is roughly quasi-isometric to a convex subset of hyperbolic space. If one is allowed to rescale the metric of X by some positive constant, then there is an embedding where distances are distorted by at most an additive constant. Another embedding theorem states that any 8-hyperbolic met-ric space embeds isometrically into a complete geodesic 8-hyperbolic space. The relation of a Gromov hyperbolic space to its boundary is fur-ther investigated. One of the applications is a characterization of the hyperbolic plane up to rough quasi-isometries.