Gromov–Hausdorff convergence

The Gromov–Hausdorff distance was introduced by David Edwards in 1975,[1][2] and it was later rediscovered and generalized by Mikhail Gromov in 1981.

If X and Y are two compact metric spaces, then dGH (X, Y) is defined to be the infimum of all numbers dH(f(X), g(Y)) for all (compact) metric spaces M and all isometric embeddings f : X → M and g : Y → M. Here dH denotes Hausdorff distance between subsets in M and the isometric embedding is understood in the global sense, i.e. it must preserve all distances, not only infinitesimally small ones; for example no compact Riemannian manifold admits such an embedding into Euclidean space of the same dimension.

A sequence (Xn, pn) of pointed metric spaces converges to a pointed metric space (Y, p) if, for each R > 0, the sequence of closed R-balls around pn in Xn converges to the closed R-ball around p in Y in the usual Gromov–Hausdorff sense.

The key ingredient in the proof was the observation that for the Cayley graph of a group with polynomial growth a sequence of rescalings converges in the pointed Gromov–Hausdorff sense.

[14] In a special case, the concept of Gromov–Hausdorff limits is closely related to large-deviations theory.