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.