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Metric spaces (cf. [[Metric space|Metric space]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q1100101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q1100102.png" />, for which there exist two mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q1100103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q1100104.png" /> and two constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q1100105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q1100106.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q1100107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q1100108.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q1100109.png" /> and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q11001010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q11001011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110010/q11001012.png" />, the following four inequalities hold:
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<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/q110010/q11001013.png" /></td> </tr></table>
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<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/q110010/q11001014.png" /></td> </tr></table>
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Metric spaces (cf. [[Metric space|Metric space]])  $  ( X,d _ {X} ) $
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and  $  ( Y,d _ {Y} ) $,
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for which there exist two mappings  $  f : X \rightarrow Y $
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and  $  g : Y \rightarrow X $
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and two constants  $  k \geq  0 $
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and  $  \lambda > 0 $
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such that for all  $  x $
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and  $  x  ^  \prime  $
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in  $  X $
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and for all  $  y $
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and  $  y  ^  \prime  $
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in  $  Y $,
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the following four inequalities hold:
  
<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/q110010/q11001015.png" /></td> </tr></table>
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$$
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d _ {Y} ( f ( x ) ,f ( x  ^  \prime  ) ) \leq  \lambda  d _ {X} ( x,x  ^  \prime  ) + k,
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$$
  
<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/q110010/q11001016.png" /></td> </tr></table>
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$$
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d _ {X} ( g ( y ) ,g ( y  ^  \prime  ) ) \leq  \lambda  d _ {Y} ( y,y  ^  \prime  ) + k,
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$$
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$$
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d _ {X} ( g \circ f ( x ) ,x ) \leq  k ,
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$$
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$$
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d _ {Y} ( f \circ g ( y ) ,y ) \leq  k.
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$$
  
 
This definition appears in [[#References|[a1]]], where it is attributed to G.A. Margulis. The relation  "X is quasi-isometric to Y"  is an [[Equivalence|equivalence]] relation between metric spaces.
 
This definition appears in [[#References|[a1]]], where it is attributed to G.A. Margulis. The relation  "X is quasi-isometric to Y"  is an [[Equivalence|equivalence]] relation between metric spaces.

Latest revision as of 08:09, 6 June 2020


Metric spaces (cf. Metric space) $ ( X,d _ {X} ) $ and $ ( Y,d _ {Y} ) $, for which there exist two mappings $ f : X \rightarrow Y $ and $ g : Y \rightarrow X $ and two constants $ k \geq 0 $ and $ \lambda > 0 $ such that for all $ x $ and $ x ^ \prime $ in $ X $ and for all $ y $ and $ y ^ \prime $ in $ Y $, the following four inequalities hold:

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

$$ d _ {X} ( g ( y ) ,g ( y ^ \prime ) ) \leq \lambda d _ {Y} ( y,y ^ \prime ) + k, $$

$$ d _ {X} ( g \circ f ( x ) ,x ) \leq k , $$

$$ d _ {Y} ( f \circ g ( y ) ,y ) \leq k. $$

This definition appears in [a1], where it is attributed to G.A. Margulis. The relation "X is quasi-isometric to Y" is an equivalence relation between metric spaces.

See also Quasi-isometry.

References

[a1] E. Ghys, "Les groupes hyperboliques" Astérisque , 189–190 (1990) pp. 203–238 (Sém. Bourbaki Exp. 722)
How to Cite This Entry:
Quasi-isometric spaces. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-isometric_spaces&oldid=14441
This article was adapted from an original article by A. Papadopoulos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article