Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [Ab] William Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Mathematics, vol. 820, Springer, Berlin, 1980. MR 590044
  • [Ahl] Lars V. Ahlfors, Lectures on quasiconformal mappings, Manuscript prepared with the assistance of Clifford J. Earle, Jr. Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. MR 0200442
  • [Ag] Stephen Agard, Remarks on the boundary mapping for a Fuchsian group, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 1–13. MR 802463, 10.5186/aasfm.1985.1004
  • [Can] James W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984), no. 2, 123–148. MR 758901, 10.1007/BF00146825
  • [Ep] D. B. A. Epstein, Curves on 2-manifolds and isotopies, Acta Math. 115 (1966), 83–107. MR 0214087
  • [Mos1] G. D. Mostow, Quasi-conformal mappings in 𝑛-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53–104. MR 0236383
  • [Mos2] -, Strong rigidity of locally symmetric spaces, Ann. of Math. Stud., no. 78, Princeton Univ. Press, Princeton, NJ, 1974.
  • [R] H. M. Reimann, Functions of bounded mean oscillation and quasiconformal mappings, Comment. Math. Helv. 49 (1974), 260–276. MR 0361067
  • [Th] W. P. Thurston, Notes on the Orbifold theorem, Princeton course notes, 1983.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53C23, 30C65, 30F40, 57M99

Retrieve articles in all journals with MSC: 53C23, 30C65, 30F40, 57M99

Additional Information

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