Quasi-isometry

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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 $A\geq 1$ , $B\geq 0$ , and $C\geq 0$ such that the following two properties both hold:

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:
$\forall x,y\in M_{1}:{\frac {1}{A}}\;d_{1}(x,y)-B\leq d_{2}(f(x),f(y))\leq A\;d_{1}(x,y)+B.$
Every point of $M_{2}$ is within the constant distance $C$ of an image point. More formally:
$\forall z\in M_{2}:\exists x\in M_{1}:d_{2}(z,f(x))\leq C.$

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.

References

Bridson, Martin R. (2008), "Geometric and combinatorial group theory", in Gowers, Timothy; Barrow-Green, June; Leader, Imre, The Princeton Companion to Mathematics, Princeton University Press, pp. 431–448, ISBN 978-0-691-11880-2

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