Quasi-isometries of hyperbolic space are almost isometries

Abstract:

In this paper we show that for $n \geq 3$ a quasi-isometry of hyperbolic n-space ${\mathbb {H}^n}$ to itself is almost an isometry, in the sense that the image of most points on a sphere of radius r are close to a sphere of the same radius. To be more precise, the result is that given $K > 1$ and $\epsilon > 0$ there is a $\delta > 0$ such that the image of any sphere of any radius r under any K-quasi-isometry lies within a distance of $\delta$ of another sphere of radius r, except for the image of a proportion $\epsilon$ of the source sphere. We illustrate our result with a quasi-isometry of ${\mathbb {H}^n}$ for which the image of a sphere is the analog of an ellipsoid in Euclidean space. There is no corresponding result when $n = 2$. This failure is illustrated by lifting to the universal cover a surface diffeomorphism which is not isotopic to an isometry.