Abstract
Almost-isometries are quasi-isometries with multiplica- tive constant one. Lifting a pair of metrics on a compact space gives quasi-isometricmetrics on the universal cover. Under someadditional hypotheses on the metrics, we show that there is no almost-isometry between the universal covers. We also show that Riemannian mani- folds that are almost-isometric have the same volume growth entropy. We then establish various rigidity results as applications.
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