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 Gromov's geometric group theory.
Suppose that is a (not necessarily continuous) function from one metric space to a second metric space . Then is called a quasi-isometry from to if there exist constants , , and such that the following two properties both hold:
- For every two points and in , the distance between their images is (up to the additive constant ) within a factor of of their original distance. More formally:
- Every point of is within the constant distance of an image point. More formally:
The two metric spaces and are called quasi-isometric if there exists a quasi-isometry from to .
The map (both with the Euclidean metric) that sends every -tuple of integers to itself is a quasi-isometry: distances are preserved exactly, and every real tuple is within distance 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 of it, so rounding changes the distance between pairs of points by adding or subtracting at most .
Every pair of finite or bounded metric spaces is quasi-isometric. In this case, every function from one space to the other is a quasi-isometry.
If is a quasi-isometry, then there exists a quasi-isometry . Indeed, may be defined by letting be any point in the image of that is within distance of , and letting be any point in .
Since the identity map is a quasi-isometry, and the composition of two quasi-isometries is a quasi-isometry, it follows that the relation of being quasi-isometric is an equivalence relation on the class of metric spaces.
Use in geometric group theory
Given a finite generating set S of a finitely generated group G, we can form the corresponding Cayley graph of S and G. This graph becomes a metric space if we declare the length of each edge to be 1. Taking a different finite generating set T results in a different graph and a different metric space, however the two spaces are quasi-isometric. This quasi-isometry class is thus an invariant of the group G. Any property of metric spaces that only depends on a space's quasi-isometry class immediately yields another invariant of groups, opening the field of group theory to geometric methods.