View source for Quasi-isometry

A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q1100201.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q1100202.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q1100203.png" /> are metric spaces (cf. [[Metric space|Metric space]]), for which there exist two constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q1100204.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q1100205.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q1100206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q1100207.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q1100208.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q1100209.png" /></td> </tr></table>

This property expresses the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q11002010.png" /> is  "Lipschitz in the large"  (cf. also [[Lipschitz constant|Lipschitz constant]]). Such a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q11002011.png" /> is also called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q11002013.png" />-quasi-isometry.

Note that this definition, which is commonly used now (see [[#References|[a1]]], §7.2.G), does not imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q11002014.png" /> is continuous. In [[#References|[a4]]], §5.9, W. Thurston considers mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q11002015.png" /> satisfying the property above but with right-hand side replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q11002016.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q11002017.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q11002018.png" />-dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q11002019.png" />, for some real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q11002020.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110020/q11002021.png" />.

====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>