# 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 Gromov's geometric group theory.

This lattice is quasi-isometric to the plane.

## Definition

Suppose that $f$ is a (not necessarily continuous) function from one metric space $(M_1,d_1)$ to a second metric space $(M_2,d_2)$. Then $f$ is called a quasi-isometry from $(M_1,d_1)$ to $(M_2,d_2)$ if there exist constants $A\ge 1$, $B\ge 0$, and $C\ge 0$ such that the following two properties both hold:

1. For every two points $x$ and $y$ in $M_1$, the distance between their images is (up to the additive constant $B$) within a factor of $A$ of their original distance. More formally:
$\forall x,y\in M_1: \frac{1}{A}\; d_1(x,y)-B\leq d_2(f(x),f(y))\leq A\; d_1(x,y)+B.$
2. Every point of $M_2$ is within the constant distance $C$ of an image point. More formally:
$\forall z\in M_2:\exists x\in M_1: d_2(z,f(x))\le C.$

The two metric spaces $(M_1,d_1)$ and $(M_2,d_2)$ are called quasi-isometric if there exists a quasi-isometry $f$ from $(M_1,d_1)$ to $(M_2,d_2)$.

## Examples

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 $\sqrt 2$.

The map $f:\mathbb{Z}^n\mapsto\mathbb{R}^n$ (both with the Euclidean metric) that sends every $n$-tuple of integers to itself is a quasi-isometry: distances are preserved exactly, and every real tuple is within distance $\sqrt{n/4}$ 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 $\sqrt{n/4}$ of it, so rounding changes the distance between pairs of points by adding or subtracting at most $2\sqrt{n/4}$.

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.

## Equivalence relation

If $f:M_1\mapsto M_2$ is a quasi-isometry, then there exists a quasi-isometry $g:M_2\mapsto M_1$. Indeed, $g(x)$ may be defined by letting $y$ be any point in the image of $f$ that is within distance $C$ of $x$, and letting $g(x)$ be any point in $f^{-1}(y)$.

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.