# Quasi-isometry

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A mapping , where and are metric spaces (cf. Metric space), for which there exist two constants and such that for all and in :

This property expresses the fact that is "Lipschitz in the large" (cf. also Lipschitz constant). Such a mapping is also called a -quasi-isometry.

Note that this definition, which is commonly used now (see [a1], §7.2.G), does not imply that is continuous. In [a4], §5.9, W. Thurston considers mappings satisfying the property above but with right-hand side replaced by . Such a mapping is continuous (cf. Continuous mapping) and Thurston calls it a pseudo-isometry. Some authors (see, e.g., [a3]) use the word quasi-isometry to denote a mapping having the property above, with the further condition that the image is -dense in , for some real number .

The importance of quasi-isometries has been fully realized in the proof of Mostow's rigidity theorem [a2]. Thurston's lectures [a4] contain an excellent exposition of this theorem for manifolds of constant curvature .

#### References

 [a1] M. Gromov, "Hyperbolic groups" S.M. Gersten (ed.) , Essays in Group Theory , MSRI Publ. , 8 , Springer (1987) pp. 75–263 [a2] G.D. Mostow, "Quasi-conformal mappings in -space and the strong rigidity of space-form" IHES Publ. Math. , 34 (1968) pp. 53–104 [a3] P. Pansu, "Métriques de Carnot–Carathéodory et quasiisométries des espaces symétriques de rang un" Ann. of Math. , 129 : 1 (1989) pp. 1–61 [a4] W. Thurston, "The geometry and topology of 3-manifolds" , Lecture Notes , Princeton Univ. Press (1976)
How to Cite This Entry:
Quasi-isometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-isometry&oldid=18488
This article was adapted from an original article by A. Papadopoulos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article