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)