a quasi-isometry
Imagine looking at two maps of the same region drawn at different scales and with some sloppiness — distances are stretched by a bounded factor and fudged by a bounded amount, but the overall layout is the same. A quasi-isometry is precisely this kind of 'same from far away' comparison between metric spaces. It deliberately ignores all local, small-scale detail and asks only whether two spaces look alike at large scale. It is the central equivalence relation of geometric group theory.
A map f: X -> Y between metric spaces is a quasi-isometry if there are constants L >= 1 and C >= 0 such that two conditions hold. First, distances are preserved up to the multiplicative L and additive C: for all x, x', (1/L) d_X(x, x') - C <= d_Y(f(x), f(x')) <= L d_X(x, x') + C. Second, f is coarsely surjective: every point of Y is within distance C of the image of f. Note f need not be continuous, injective, or surjective — it only respects distances up to bounded multiplicative and additive errors and nearly covers the target. Two spaces are quasi-isometric if such an f exists; this is an equivalence relation. The simplest example: the integers Z and the real line R are quasi-isometric, with the inclusion Z -> R as the map (L = 1, C = 1), even though one is discrete and the other a continuum — at large scale they are the same.
Quasi-isometry is the right lens for groups because the word metric is only defined up to a choice of generators, and that choice changes the metric exactly within a quasi-isometry. So any property of a group preserved by quasi-isometry — being hyperbolic, growth type, number of ends, being virtually nilpotent, amenability — is a genuine algebraic invariant readable from the geometry. These are the quasi-isometry invariants, the real content of the field. The big caveat: quasi-isometry is blind to anything local or finite. Finite groups are all quasi-isometric to a point; passing to a finite-index subgroup, or quotienting by a finite normal subgroup, does not change the quasi-isometry class. So quasi-isometry detects only the large-scale shape, and two very different-looking groups can be quasi-isometric (the Svarc-Milnor lemma exploits exactly this).
The map f: Z -> R given by inclusion is a quasi-isometry: distances are preserved exactly (L = 1, C = 0 on the multiplicative side) and every real number is within 1/2 of an integer, so the image is coarsely dense (C = 1/2 suffices for surjectivity). At small scale Z and R are utterly different — one is a discrete set of points, the other a continuum with no gaps — but at large scale they are indistinguishable. This is why a finitely generated group and any space it acts on geometrically have the same large-scale geometry.
Z and R are quasi-isometric: identical at large scale despite discrete versus continuous local structure.
Quasi-isometries need not be continuous, injective, or surjective, and they erase all finite and local information. Any two finite metric spaces are quasi-isometric to a point, and finite-index subgroups share a group's quasi-isometry type — never expect quasi-isometry to see small-scale structure.