# Quasi-isometry

A mapping $f : X \rightarrow Y$, where $( X,d _ {X} )$ and $( Y,d _ {Y} )$ are metric spaces (cf. Metric space), for which there exist two constants $\lambda > 0$ and $k \geq 0$ such that for all $x$ and $x ^ \prime$ in $X$:

$$\lambda ^ {- 1 } d _ {X} ( x,x ^ \prime ) - k \leq d _ {Y} ( f ( x ) ,f ( x ^ \prime ) ) \leq \lambda d _ {X} ( x,x ^ \prime ) + k.$$

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

Note that this definition, which is commonly used now (see [a1], §7.2.G), does not imply that $f$ is continuous. In [a4], §5.9, W. Thurston considers mappings $f$ satisfying the property above but with right-hand side replaced by $\lambda d _ {X} ( x,x ^ \prime )$. 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 $f ( X )$ is $\delta$- dense in $Y$, for some real number $\delta$.

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 $- 1$.

#### 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=48387
This article was adapted from an original article by A. Papadopoulos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article