Amenability, bilipschitz equivalence, and the von Neumann conjecture

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 we finite homology which we will apply to manifolds in a later paper. In his fundamental paper on amenability, Von Neumann conjectured that if Γ is a finitely generated, non-amenable group, then Γ has a subgroup which is free on two generators. While this conjecture proved to be false in general [O], it is true for many classes of groups. Given the recent interest in studying groups via geometric methods, it seems natural to ask for a geometric version of the conjecture. If Γ does contain a free subgroup then its cosets will partition Γ into copies of the free group. The free group is, geometrically, the regular 4-valent tree. Our " geometric Von Neumann conjecture " , is: Theorem 1 If X is a uniformly discrete space of bounded geometry (in particular, for X a finitely generated group or a net in a leaf of a foliation of a compact manifold), X is non-amenable iff it admits a partition with pieces uniformly bilipschitz equivalent to the regular 4-valent tree. The proof depends on constructing a bilipschitz eqivalence of X × {0, 1} and X near the projection map. Our main technical result is a general 1