Gromov-Hausdorff convergence

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Gromov-Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence.

1 Gromov-Hausdorff distance
2 Pointed Gromov-Hausdorff convergence
3 Applications
4 References

[edit] Gromov-Hausdorff distance

Gromov-Hausdorff distance measures how far two compact metric spaces are from being isometric. If X and Y are two compact metric spaces, then d_GH (X,Y ) is defined to be the infimum of all numbers d_H(f (X ), g (Y )) for all metric spaces M and all isometric embeddings f :X→M and g :Y→M.

(Here d_H 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 of negative sectional curvature admits such an embedding into Euclidean space.)

The Gromov-Hausdorff distance turns the set of all isometry classes of compact metric spaces into a metric space, and it therefore defines a notion of convergence for sequences of compact metric spaces, called Gromov-Hausdorff convergence.

[edit] Pointed Gromov-Hausdorff convergence

Pointed Gromov-Hausdorff convergence is an appropriate analog of Gromov-Hausdorff convergence for non-compact spaces.

Given a sequence (X_n, p_n) of locally compact complete length metric spaces with distinguished points, it converges to (Y,p) if for any R > 0 the closed R-balls around p_n in X_n converge to the closed R-ball around p in Y in the usual Gromov-Hausdorff sense.

[edit] Applications

The notion of Gromov-Hausdorff convergence was first used by Gromov to prove that any discrete group with polynomial growth is almost nilpotent (i.e. it contains a nilpotent subgroup of finite index). See Gromov's theorem on groups of polynomial growth. The key ingredient in the proof was the almost trivial observation that for the Cayley graph of a group with polynomial growth a sequence of rescalings converges in the pointed Gromov-Hausdorff sense.

Another simple and very useful result in Riemannian geometry is Gromov's compactness theorem, which states that the set of Riemannian manifolds with Ricci curvature ≥c and diameter ≤D is pre-compact in the Gromov-Hausdorff metric.