  Almost-isometries are quasi-isometries with multiplicative constant one.
Lifting a pair of metrics on a compact space gives quasi-isometric metrics on
the universal cover. Under some additional hypotheses on the metrics, we show
that there is no almost-isometry between the universal covers. We show that
Riemannian manifolds which are almost-isometric have the same volume growth
entropy. We establish various rigidity results as applications.
