# 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.