# Quasi-isometric spaces

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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=48386
This article was adapted from an original article by A. Papadopoulos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article