# GromovHausdorff convergence

**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 *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*

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*M*and

*g*:

*Y*

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*M*.

(Here *d*_{H} 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 (*X _{i},p_{i}*) 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

*p*in

_{i}*X*converges to the

_{i}*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 curvature ≥*c* and diameter ≤*D* is pre-compact in the Gromov-Hausdorff metric.