, Volume 10, Issue 2, pp 266-306

Embeddings of Gromov hyperbolic spaces

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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 \( \delta \) -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.

Submitted: October 1998, Revised version: January 1999.