Gromov-Hausdorff convergence

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Gromov-Hausdorff convergence is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence.

Table of contents

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

Gromov-Hausdorff distance

Gromov-Hausdorff distance measures how two compact metric spaces are far from being isometric. Let X and Y be two compact metric spaces, and then d_GH(X,Y) is the minimum 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.

(The isometric embedding here 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.

Let (X_i,p_i) be a sequence of locally compact complete length metric spaces with marked points, it is converging to (Y,p) if for any R>0 the closed R-balls around p_i in X_i is converging 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). 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 curvature ≥c and diameter ≤D is pre-compact in the Gromov-Hausdorff metric.