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I am looking for applications of the notion of Gromov-Hausdorff convergence to prove theorems that a priori have nothing to do with it. Examples that I am aware of (thanks to wikipedia and google):

What are more examples? Ideally they would be along the lines of Gromov's theorem, or proofs of geometric facts, but I'm interested to hear about anything.

As a subquestion, are there interesting applications of Gromov's compactness theorem to prove results about manifolds with bounded Ricci which have nothing to do with GH convergence?

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One of the original uses of Gromov-Hausdorff distance in Riemannian geometry was to prove finiteness and pinching theorems. This is described, for example, in Peter Petersen's textbook on Riemannian geometry. –  Deane Yang Feb 28 '11 at 0:16
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Gromov's Theorem was, as far as I'm aware, the first but very far from the last application of Gromov--Hausdorff distance to group theory. One particularly fruitful line of reasoning starts with a sequence of actions of a group $\Gamma$ on (Gromov)-hyperbolic metric spaces. In the (rescaled) limit, one gets an action on an $\mathbb{R}$-tree, to which one can apply Rips' structure theory.

Examples include:

  • Bestvina's construction of a boundary for the set of hyperbolic structures on a manifold;
  • Paulin's Theorem: if $\Gamma$ is word-hyperbolic and $\mathrm{Out}(\Gamma)$ is infinite then $\Gamma$ splits over a virtually cyclic subgroup;
  • Sela's theorem that word-hyperbolic groups are Hopfian;
  • this idea is the principal tool in Sela's solution to Tarski's Problem on the elementary theory of free groups;
  • etc.
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@Henry: These examples employ asymptotic cones, not GH convergence. GH limit of a sequence of metric spaces exists only if the sequence of spaces is uniformly locally compact which is true for groups of polynomial growth and not true for trees of degree at least 3. I have made the same mistake several times also before several people pointed that out to me. –  Mark Sapir Feb 27 '11 at 13:31
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Mark, actually these examples employ the equivariant Gromov--Hausdorff topology. You can also use the pointed Gromov--Hausdorff topology. I agree that this isn't strictly speaking the usual Gromov--Hausdorff topology, but nonetheless I felt that they were in the spirit of the question. Asymptotic cones are one way of proving that a limit exists, but not the only way. –  HJRW Feb 27 '11 at 15:50
    
What is equivariant Gromov-Hausdorff convergence? –  Otis Chodosh Feb 28 '11 at 9:29
    
The definition is given in various papers of Paulin. This is probably the place to look: Paulin, Frédéric, Topologie de Gromov équivariante, structures hyperboliques et arbres réels. (French) [Equivariant Gromov topology, hyperbolic structures and real trees] Invent. Math. 94 (1988), no. 1, 53–80. –  HJRW Feb 28 '11 at 14:59
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Gromov-Hausdorff convergence (with basepoints) is used quite a bit in the classification of Kleinian groups. In particular, it is used to resolve the Ahlfors measure conjecture and the ending lamination conjecture. It is also used in the proof of the geometrization conjecture (and as a corollary the Poincare conjecture).

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Many people have applied Gromov-Hausdorff convergence to obtain information about Riemannian manifolds with nonnegative Ricci curvature. Controls on Gromov-Hausdorff limits can lead to controls on diameters. For example in my thesis I proved, among other things, that manifolds with nonnegative Ricci curvature and linear volume growth have sublinear diameter growth. The proof uses Gromov-Hausdorff convergence and methods of Cheeger-Colding to obtain that convergence.

Other times Riemannian geometers use Gromov-Hausdorff convergence to come up with an idea but later simplify the proof in a way which circumvents actually mentioning the Gromov-Hausdorff convergence. I have a paper about fundamental groups of manifolds with nonnegative Ricci curvature and my original proof of the main theorem involved taking a Gromov-Hausdorff limit (as described in the final section of the paper). Then I thought of a simplification which allows one to obtain the main theorem without appealing to Gromov's compactness theorem.

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The scaling limits of several families of random graphs are shown to exist by using the idea of Gromov-Hausdorff convergence to certain random metric spaces.

For instance, uniformly chosen triangulations of the sphere with $n$ faces endowed with the graph distance have been proved to converge (in the Gromov-Hausdorff sense) after rescaling distances by $n^{-1/4}$ to a particular random metric space called the Brownian map. See the references in this earlier answer of mine.

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Another graph-theoretic application.

Given an undirected finite graph $G$, the Colin de Verdière graph invariant $\mu(G)$ is defined as the maximum multiplicity of the second smallest eigenvalue of a matrix $M\in O_G$, provided that this eigenvalue is structurally stable. Here, $O_G$ is an interesting (for $G$) subset of the symmetric matrices called the Schrödinger-like operators on $G$.

It turns out that this invariant is very nice in the following sense: if $H$ is a minor of $G$, then the invariant verifies $\mu(H) \leq\mu(G)$. A trick used to prove this result is to put weights on the edges of $G$ to turn it into a metric space (with the obvious distance of smallest sum of weight in a path). Then, any minor $H$ can be obtained as a Gromov-Hausdorff limit of $G$ with suitable weights (contracted edges see their weight go to zero and deleted edges have their weight go to infinity), and conversely, any sequence of weights on the edges of $G$ that converges gives a minor.

Now that I wrote it down, it does not seem like such a powerful remark, but, as I recall, this shift in point of view is (at the very least) very convenient to prove the result.

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Facundo Memoli applied Gromov-Hausdorff distance to shape matching in his Ph.D. thesis.

http://math.stanford.edu/~memoli/research.html

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Stephen Keith used the Gromov-Hausdorff convergence to study the existence of (measurable) differentiable structure on metric measure spaces that supports a Poincare inequality or K-Lip-lip condition.

Juha Heinonen, Jeff Cheeger and Stephen keith also used this method as a standard blow up argument in related questions.

Heinonen, Juha; Keith, Stephen Flat forms, bi-Lipschitz parameterizations, and smoothability of manifolds. Publ. Math. Inst. Hautes Études Sci. No. 113 (2011), 1–37.

Keith, Stephen A differentiable structure for metric measure spaces. Adv. Math. 183 (2004), no. 2, 271–315.

J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (3) (1999) 428–517.

J. Cheeger, T.H. Colding, On the structure of spaces with Ricci curvature bounded below. I, II, III, J. Differential Geom.

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