View source for Quasi-isometry

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A mapping  $  f : X \rightarrow Y $, 
where  $  ( X,d _ {X} ) $
and  $  ( Y,d _ {Y} ) $
are metric spaces (cf. [[Metric space|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|Lipschitz constant]]). Such a mapping  $  f $
is also called a  $  ( \lambda,k ) $-
quasi-isometry.

Note that this definition, which is commonly used now (see [[#References|[a1]]], §7.2.G), does not imply that  $  f $
is continuous. In [[#References|[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|Continuous mapping]]) and Thurston calls it a pseudo-isometry. Some authors (see, e.g., [[#References|[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 [[#References|[a2]]]. Thurston's lectures [[#References|[a4]]] contain an excellent exposition of this theorem for manifolds of constant curvature  $  - 1 $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Gromov,   "Hyperbolic groups"  S.M. Gersten (ed.) , ''Essays in Group Theory'' , ''MSRI Publ.'' , '''8''' , Springer  (1987)  pp. 75–263</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.D. Mostow,   "Quasi-conformal mappings in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q11002022.png" />-space and the strong rigidity of space-form"  ''IHES Publ. Math.'' , '''34'''  (1968)  pp. 53–104</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Pansu,   "Métriques de Carnot–Carathéodory et quasiisométries des espaces symétriques de rang un"  ''Ann. of Math.'' , '''129''' :  1  (1989)  pp. 1–61</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W. Thurston,   "The geometry and topology of 3-manifolds" , ''Lecture Notes'' , Princeton Univ. Press  (1976)</TD></TR></table>