quasiisometries

Quasiisometries are maps between metric spaces that preserve large-scale geometry up to controlled distortion. They are central in geometric group theory, where spaces are studied up to coarse equivalence rather than exact isometry.

Definition. Let (X, dX) and (Y, dY) be metric spaces. A map f: X → Y is a (K, C)-quasi-isometry if there exist constants K ≥ 1 and C ≥ 0 such that for all x, x′ in X,

(1/K) dX(x, x′) − C ≤ dY(f(x), f(x′)) ≤ K dX(x, x′) + C,

and every point y in Y lies within distance C of some f(x) (i.e., Y is contained in the C-neighborhood of f(X)). Some authors require a bound with a possibly different constant for coarse surjectivity, or permit a quasi-inverse g: Y → X with similar inequalities.

Two spaces are quasi-isometric if there exists a quasi-isometry X → Y and a quasi-isometry Y → X that are coarse inverses of each other (their compositions are close to the respective identities).

Key properties. Quasi-isometries are stable under composition; they induce an equivalence relation on classes of metric spaces. They do not preserve small-scale structure but preserve the large-scale, or coarse, geometry. Consequently, many global features are invariants under quasi-isometry, such as growth type of balls, asymptotic dimension, and, in hyperbolic spaces, the Gromov boundary. In geometric group theory, finitely generated groups equipped with word metrics are studied up to quasi-isometry; quasi-isometric groups share many large-scale properties.

Examples. The inclusion Z → R is a quasi-isometry since the image is dense in R at a fixed bound. The map n ↦ 2n on Z is a quasi-isometry from Z to Z, illustrating distortion-free large-scale structure despite non-surjectivity.

History. The notion was developed in the late 20th century, notably by M. Gromov, and has since become foundational for understanding spaces and groups through their large-scale geometry.