© 1996, Topology Atlas

Coarse geometry is the study of metric spaces (or perhaps more general objects) from a `large scale' point of view, so that two space which `look the same from a great distance' are actually equivalent.

More precisely, let X and Y be metric spaces. A map (not necessarily continuous) f from X to Y is a coarse equivalence if there are constants C, A such that d(x,y) is less than C d(f(x),f(y)) + A and d(f(x),f(y)) is less than C d(x,y) + A, for all x and y in X.

Examples: Any compact space is coarse equivalent to a point. R is coarse equivalent to Z. The universal cover of a compact manifold is coarse equivalent to the fundamental group.

One studies the subject because in various contexts it is in fact true that the relevant geometric properties of metric spaces are the coarse ones. For instance, the beautiful notion of hyperbolic group, due to Gromov, refers to a group which is `coarsely negatively curved'. The coarse geometry of negatively curved spaces was exploited by Mostow in his proof of the rigidity theorem (Strong Rigidity for Locally Symmetric Spaces, Princeton, 1973). The Novikov and Borel conjectures, closely related to rigidity theory, seem to involve the coarse geometry of infinite groups. And so on. Here is an inspirational quotation from Gromov (Asymptotic Invariants of Infinite Groups, Cambridge, 1993).

A group G with a given system of generators carries a unique maximal left invariant distance function for which the distance from each generator and its inverse to the identity is 1. This distance function, called the word metric associated to the given system of generators, makes G as subject to a geometric scrutiny as any other metric space.

This space may appear boring and uneventful to a geometer's eye since it is discrete and the traditional (e.g. topological and infinitesimal) machinery does not run in G. To regain the geometric perspective one has to change one's position and move the observation point far away from G. Then the metric in G seen from the distance d becomes the original distance divided by d and as d tends to infinity the points in G coalesce into a connected continuous solid unity which occupies the visual horizon without any gaps or holes and fills our geometers heart with joy....

One may start to feel uncomfortable by realizing how much structure has been lost as one passed from G to the quasi-isometry class of G with its word metric. Indeed, one barters here the rigid crystalline beauty of a group for a soft and flabby chunk of geometry where all measurements have built-in errors. But something amazing and unexpected happens here as was discovered by Mostow in 1968: the quasi-isometric (or large-scale) geometry turns out to be far more rich and powerful than appears at first sight. In fact one believes nowadays that most essential elements of an infinite group are quasi-isometry invariant.