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.

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:

The two metric spaces ${\displaystyle (M_{1},d_{1})}$ and ${\displaystyle (M_{2},d_{2})}$ are called quasi-isometric if there exists a quasi-isometry ${\displaystyle f}$ from ${\displaystyle (M_{1},d_{1})}$ to ${\displaystyle (M_{2},d_{2})}$.

A map is called a quasi-isometric embedding if it satisfies the first condition but not necessarily the second (i.e. it is coarsely Lipschitz but may fail to be coarsely surjective). In other words, if through the map, ${\displaystyle (M_{1},d_{1})}$ is quasi-isometric to a subspace of ${\displaystyle (M_{2},d_{2})}$.

Two metric spaces M₁ and M₂ are said to be quasi-isometric, denoted ${\displaystyle M_{1}{\underset {q.i.}{\sim }}M_{2}}$, if there exists a quasi-isometry ${\displaystyle f:M_{1}\to M_{2}}$.

The map between the Euclidean plane and the plane with the Manhattan distance that sends every point to itself is a quasi-isometry: in it, distances are multiplied by a factor of at most ${\displaystyle {\sqrt {2}}}$. Note that there can be no isometry, since, for example, the points ${\displaystyle (1,0),(-1,0),(0,1),(0,-1)}$ are of equal distance to each other in Manhattan distance, but in the Euclidean plane, there are no 4 points that are of equal distance to each other.

The map ${\displaystyle f:\mathbb {Z} ^{n}\to \mathbb {R} ^{n}}$ (both with the Euclidean metric) that sends every ${\displaystyle n}$-tuple of integers to itself is a quasi-isometry: distances are preserved exactly, and every real tuple is within distance ${\displaystyle {\sqrt {n/4}}}$ of an integer tuple. In the other direction, the discontinuous function that rounds every tuple of real numbers to the nearest integer tuple is also a quasi-isometry: each point is taken by this map to a point within distance ${\displaystyle {\sqrt {n/4}}}$ of it, so rounding changes the distance between pairs of points by adding or subtracting at most ${\displaystyle 2{\sqrt {n/4}}}$.