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 conjectureending lamination conjecture. It is also used in the proof of the geometrization conjecturegeometrization conjecture (and as a corollary the Poincare conjecture).
Martin Sleziak
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