quasi-isometry
Imagine viewing two maps from very far away, or with blurry glasses that ignore details up to some fixed size. A city block and the same block with a few extra alleys look identical from orbit; an exact metre is invisible. A quasi-isometry formalizes ‘the same shape on the large scale’: it is a map between metric spaces that may stretch, squish, and jiggle distances, but only by a bounded linear factor plus a bounded additive error.
A map f from a metric space X to a metric space Y is a (K, C)-quasi-isometric embedding if for all points x, x' one has (1/K)d(x, x') - C ≤ d(f(x), f(x')) ≤ K d(x, x') + C. It is a quasi-isometry if, in addition, its image is C-dense: every point of Y lies within C of some f(x). Quasi-isometry is an equivalence relation on metric spaces, coarser than bi-Lipschitz (it tolerates additive error and need not be continuous or even well-defined pointwise after coarsening).
The decisive theorem of Schwarz and Milnor says that if a group G acts properly discontinuously and cocompactly by isometries on a proper geodesic metric space X, then G with any word metric is quasi-isometric to X. Consequently every finitely generated group has a well-defined quasi-isometry type, and quasi-isometry invariants — growth, ends, hyperbolicity, finite presentability, virtual nilpotence — are genuine invariants of the group. Geometric group theory is in large part the study of groups up to quasi-isometry.
The integers Z embed in the real line R, and the inclusion is a quasi-isometry: every real number is within 1/2 of an integer, and distances agree exactly. So the discrete group Z and the continuous space R have the same large-scale geometry.
Z and R are quasi-isometric: the gaps between integers are invisible at large scale.
Gromov's polynomial growth theorem is a landmark quasi-isometry rigidity result: a finitely generated group quasi-isometric to one of polynomial growth is itself virtually nilpotent.