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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.

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Submitted: October 1998, Revised version: January 1999.

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Bonk, M., Schramm, O. Embeddings of Gromov hyperbolic spaces . GAFA, Geom. funct. anal. 10, 266–306 (2000). https://doi.org/10.1007/s000390050009

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  • DOI: https://doi.org/10.1007/s000390050009

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