  We determine when a quasi-isometry between discrete spaces is at bounded
distance from a bilipschitz map. From this we prove a geometric version of the
Von Neumann conjecture on amenability. We also get some examples in geometric
groups theory which show that the sign of the Euler characteristic is not a
coarse invariant. Finally we get some general results on uniformly finite
homology which we will apply to manifolds in a later paper.
