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)