Embeddings of Gromov hyperbolic spaces
- Cite this article as:
- Bonk, M. & Schramm, O. GAFA, Geom. funct. anal. (2000) 10: 266. doi:10.1007/s000390050009
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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 \( \delta \)-hyperbolic metric space embeds isometrically into a complete geodesic \( \delta \)-hyperbolic space.¶The relation of a Gromov hyperbolic space to its boundary is further investigated. One of the applications is a characterization of the hyperbolic plane up to rough quasi-isometries.