Gromov–Hausdorff convergence

Gromov–Hausdorff convergence

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.

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 "dGH" ("X,Y" )is defined to be the infimum of all numbers "dH"("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. A manifold to which such a sequence converges is called the Hausdorff limit of the sequence.

Pointed Gromov–Hausdorff convergence

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

Given a sequence ("Xn, pn") 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 "pn" in "Xn" converge to the closed "R"-ball around "p" in "Y" in the usual Gromov–Hausdorff sense.

Applications

The notion of Gromov–Hausdorff convergence was first used by Gromov to prove thatany 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 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 relatively compact in the Gromov–Hausdorff metric.

References

* M. Gromov "Structures métriques pour les variétés riemanniennes", edited by Lafontaine and Pierre Pansu, 1980
* M. Gromov. "Metric structures for Riemannian and non-Riemannian spaces", Birkhäuser (1999). ISBN 0-8176-3898-9. (translation with additional content)
* Burago-Burago-Ivanov "A Course in Metric Geometry", AMS GSM 33 (readable by first year graduate students)


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