# Quasi-isometric spaces

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