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 a quasi-isometry of hyperbolic *n*-space 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 and there is a such that the image of any sphere of any radius *r* under any *K*-quasi-isometry lies within a distance of of another sphere of radius *r*, except for the image of a proportion of the source sphere. We illustrate our result with a quasi-isometry of for which the image of a sphere is the analog of an ellipsoid in Euclidean space. There is no corresponding result when . This failure is illustrated by lifting to the universal cover a surface diffeomorphism which is not isotopic to an isometry.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1995-1307505-8

Keywords:
Quasi-isometry,
hyperbolic space,
Mostow rigidity

Article copyright:
© Copyright 1995
American Mathematical Society