Quasi-isometry
A mapping , where
and
are metric spaces (cf. Metric space), for which there exist two constants
and
such that for all
and
in
:
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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 ![]() |
[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) |
Quasi-isometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-isometry&oldid=18488