- Gromov–Hausdorff convergence
Gromov–Hausdorff convergence, named after
Mikhail Gromov andFelix Hausdorff , is a notion for convergence ofmetric space s 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 compactRiemannian manifold of negativesectional curvature admits such an embedding intoEuclidean 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
sequence s 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). SeeGromov's theorem on groups of polynomial growth .The key ingredient in the proof was the observation that for theCayley 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 withRicci curvature ≥"c" anddiameter ≤"D" isrelatively 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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