Quasi-isometry

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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[edit]

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[edit]

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[edit]

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.

References[edit]