Information AboutGromov-hausdorff Convergence |
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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 minimum 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 extrinsic 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 .) 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. 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 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. REFERENCES
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