GromovHausdorff convergence

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Gromov-Hausdorff convergence 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. Let X and Y be two compact metric spaces, and then dGH (X,Y ) is the minimum of all numbers dH(f (X ), g (Y )) for all metric spaces M and all isometric embeddings f :XFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \to} M and g :YFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \to} M.

(Here dH denotes Hausdorff distance between subsets in M and the isometric embedding understood in the extrinsic sense, i.e it must preserve all distances, not only infinitesimally small, for example no compact Riemannian manifold admit such embedding into Euclidean space)

Gromov-Hausdorff distance turns the set of all isometry classes of compact metric spaces into a metric space, and what is more important into a topological space, i.e. it defines convergence for sequence of compact metric spaces which is 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 (Xi,pi) of locally compact complete length metric spaces with marked points, it converges to (Y,p) if for any R > 0 the closed R-balls around pi in Xi converges to the R-ball around p in Y in 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 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.

Yet one more simple and very useful result in Riemannian geometry is Gromov's compactness theorem, which states that the set of Riemannian manifolds with Ricci curvaturec and diameterD is pre-compact in the Gromov-Hausdorff metric.