Quasiisometry

Quasi-isometry is a coarse geometric relation between metric spaces. A map f: X → Y is a quasi-isometry if there are constants λ ≥ 1 and ε ≥ 0 such that for all x, x' in X, (1/λ) dX(x, x') − ε ≤ dY(f(x), f(x')) ≤ λ dX(x, x') + ε, and every y in Y lies within a fixed distance C of f(X). If a quasi-inverse g: Y → X exists, then X and Y are quasi-isometric.

Examples: Z and R with their usual metrics are quasi-isometric. For finitely generated groups, any two Cayley graphs with finite generating sets are quasi-isometric, so the group’s large-scale geometry is an invariant of the group rather than of the presentation.

Quasi-isometries preserve large-scale features such as the number of ends, hyperbolicity, and the growth type of finitely generated groups. They ignore small-scale details like exact distances or local curvature, allowing spaces with different micro-geometry to be equivalent at infinity.

Related ideas include quasi-isometric embeddings and coarse (Gromov) equivalence. Quasi-isometry classes organize spaces and groups by their large-scale geometry and are central to geometric group theory and Gromov's program to classify algebraic objects via geometry.