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Quasi-isometries of hyperbolic space are almost isometries

Author: Daryl Cooper
Journal: Proc. Amer. Math. Soc. 123 (1995), 2221-2227
MSC: Primary 53C23; Secondary 30C65, 30F40, 57M99
MathSciNet review: 1307505
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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.

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Keywords: Quasi-isometry, hyperbolic space, Mostow rigidity
Article copyright: © Copyright 1995 American Mathematical Society

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