In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. The property of being quasi-isometric behaves like an equivalence relation on the class of metric spaces. The concept of quasi-isometry is especially important in geometric group theory, following the work of Gromov.
Definition Suppose that f {\displaystyle f} is a (not necessarily continuous) function from one metric space ( M 1 , d 1 ) {\displaystyle (M_{1},d_{1})} to a second metric space ( M 2 , d 2 ) {\displaystyle (M_{2},d_{2})} . Then f {\displaystyle f} is called a quasi-isometry from ( M 1 , d 1 ) {\displaystyle (M_{1},d_{1})} to ( M 2 , d 2 ) {\displaystyle (M_{2},d_{2})} if there exist constants A ≥ 1 {\displaystyle A\geq 1} , B ≥ 0 {\displaystyle B\geq 0} , and C ≥ 0 {\displaystyle C\geq 0} such that the following two properties both hold:
For every two points x {\displaystyle x} and y {\displaystyle y} in M 1 {\displaystyle M_{1}} , the distance between their images is up to the additive constant B {\displaystyle B} within a factor of A {\displaystyle A} of their original distance. More formally:
∀ x , y ∈ M 1 : 1 A d 1 ( x , y ) − B ≤ d 2 ( f ( x ) , f ( y ) ) ≤ A d 1 ( x , y ) + B . {\displaystyle \forall x,y\in M_{1}:{\frac {1}{A}}\;d_{1}(x,y)-B\leq d_{2}(f(x),f(y))\leq A\;d_{1}(x,y)+B.}
Every point of M 2 {\displaystyle M_{2}} is within the constant distance C {\displaystyle C} of an image point. More formally:
∀ z ∈ M 2 : ∃ x ∈ M 1 : d 2 ( z , f ( x ) ) ≤ C . {\displaystyle \forall z\in M_{2}:\exists x\in M_{1}:d_{2}(z,f(x))\leq C.}
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