Rigidity of Almost-Isometric Universal Covers
  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. 
Aditi Kar, Jean-Francois Lafont, Benjamin Schmidt
math.GR math.DG math.GT