Quasi-isometry

In mathematics, quasi-isometry is an equivalence relation on metric spaces that ignores their small-scale details in favor of their coarse structure. The concept is especially important in geometric group theory following the work of Gromov.

Suppose that ${\displaystyle f}$ is a (not necessarily continuous) function from one metric space ${\displaystyle (M_{1},d_{1})}$ to a second metric space ${\displaystyle (M_{2},d_{2})}$. Then ${\displaystyle f}$ is called a quasi-isometry from ${\displaystyle (M_{1},d_{1})}$ to ${\displaystyle (M_{2},d_{2})}$ if there exist constants ${\displaystyle A\geq 1}$, ${\displaystyle B\geq 0}$, and ${\displaystyle C\geq 0}$ such that the following two properties both hold:

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