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.