Embeddings of Gromov hyperbolic spaces

Citation
M. Bonk et O. Schramm, Embeddings of Gromov hyperbolic spaces, GEO FUNCT A, 10(2), 2000, pp. 266-306
Citations number
14
Categorie Soggetti
Mathematics
Journal title
GEOMETRIC AND FUNCTIONAL ANALYSIS
ISSN journal
1016443X → ACNP
Volume
10
Issue
2
Year of publication
2000
Pages
266 - 306
Database
ISI
SICI code
1016-443X(2000)10:2<266:EOGHS>2.0.ZU;2-6
Abstract
It is shown that a Gromov hyperbolic geodesic metric space X with bounded g rowth at some scale is roughly quasi-isometric to a convex subset of hyperb olic space. If one is allowed to rescale the metric of X by some positive c onstant, then there is an embedding where distances are distorted by at mos t an additive constant. Another embedding theorem states that any S-hyperbolic metric space embeds isometrically into a complete geodesic S-hyperbolic space. The relation of a Gromov hyperbolic space to its boundary is further invest igated. One of the applications is a characterization of the hyperbolic pla ne up to rough quasi-isometries.