Continuation of expository series on Gromov hyperbolicity. Recall that a map is a quasi-isometry if there are constants
such that
for all
. This is a coarse version of the bi-Lipschitz condition. Surprisingly, Gromov hyperbolicity is preserved under quasi-isometries of geodesic spaces. The surprising part is that the multiplicative constant
does not kill the additive constant
.
Theorem. Suppose and
are geodesic metric spaces, and
is Gromov hyperbolic. If there exists a quasi-isometry
, then
is also Gromov hyperbolic.
Proof goes like this. Assuming that contains a fat geodesic triangle
, we consider the geodesic triangle in
with vertices
, and want to prove that it is also fat. Since
is a quasi-isometry, it follows that the images of geodesics
,
and
form a roughly-triangular shape which has the fatness property: there is a point on one of the sides that is far away from the other two sides. The problem reduces to showing that this roughly-triangular shape lies within a certain distance
(independent of
) from the actual geodesic triangle with vertices
. This is known as stability of quasi-geodesics. A quasi-geodesic is a quasi-isometric image of a line segment, similar to how a geodesic is a (locally) isometric image of a segment.
By the way, quasi-geodesic stability fails in . We can connect the points
and
by the quasi-geodesic
, which is at distance
from the true geodesic between these points.

I’ll prove a more specialized and weaker statement, which however contains the essence of the full result. Namely, let denote the hyperbolic plane and assume that
is bi-Lipschitz:
for all
. The claim is that the image of
lies in the
-neighborhood of the geodesic through
and
, where
depends only on
.
There are three standard models of hyperbolic plane: disk, halfplane and infinite strip. I’ll use the last one, because it’s the only model in which a geodesic is represented by Euclidean line. Specifically, is identified with the infinite strip
equipped with the metric
. (To see where the metric comes from, apply
to map the strip onto upper halfplane and pull back the hyperbolic metric
.)
The hyperbolic and Euclidean metrics coincide on the real line, which is where we place and
with the help of some hyperbolic isometry. Let
be our quasi-geodesic. Being a bi-Lipschitz image of a line segment,
satisfies the chord-arc condition: the length of any subarc of
does not exceed
times the distance between its endpoints. Pick
such that
. Let
be the hyperbolic distance between the lines
and
. This distance could be calculated as
, but I’d rather keep this integral as an exquisite Calculus II torture device.
The problem facing us is that quasigeodesic may be times longer than the distance between its endpoints, which seems to allows it to wander far off the straight path. However, it turns out there is a uniform bound on the length of any subarc
of
that lies within the substrip
. We lose no generality in assuming that the endpoints of
are on the line
; they will be denoted
,
. The key point is that connecting these two points within
is rather inefficient, and such inefficiency is controlled by the chord-arc property.

The hyperbolic distance between is at most
, because we can go from
to
(distance
), then from
to
(distance
), and finally from
to
(distance
). On the other hand, the length of
is at least
because the density of hyperbolic metric is at least
where
lives. The chord-arc property yields
, which simplifies to
. Hence, the distance between the endpoints of
is at most
, and another application of the chord-arc property bounds the length of
by
.
In conclusion, the claimed stability result holds with .
Complete proofs can be found in many books, for example Metric Spaces of Non-Positive Curvature by Bridson and Haefliger or Elements of Asymptotic Geometry by Buyalo and Schroeder. I used Schroeder’s lecture notes An introduction to asymptotic geometry.