Aperiodic tilings, positive scalar curvature, and amenability of spaces

The object of this paper is to begin a geometric study of noncompact spaces whose local structure has bounded complexity. Manifolds of this sort arise as leaves of foliations of compact manifolds and as their universal covers. We shall introduce a coarse homology theory using chains of bounded complexity and study some of its first properties. The most interesting result characterizes when H uf (X) vanishes as an analogue and strengthening of F0lner's amenability criterion for groups in terms of isoperimetric inequalities. (See [4].) One can view this result as producing a successful infinite Ponzi scheme on any nonamenable space. Each point, with only finite resources, gives to some of its neighbors some of these resources, yet receives more from the remaining neighbors. As one can imagine this is useful for eliminating obstructions on noncompact spaces. This has a number of applications. We present two of them. The first produces tilings that are "unbalanced" on any nonamenable polyhedron. Unbalanced tilings are automatically aperiodic and this gives many examples of sets of tiles that tile only aperiodically. Unfortunately, imbalance is a particularly unsubtle reason for aperiodicity so that the aperiodic tilings of Euclidean space (Penrose tilings) are necessarily not accessible to our method. On the other hand, most other simply connected noncompact symmetric spaces even have unbalanced tilings using our criterion. The second application regards characteristic numbers of manifolds whose universal covers have positive scalar curvature. We prove a converse to a theorem of Roe. We show that for any nonamenable group F one can find a spin manifold with fundamental group F, with nonzero A-genus whose universal cover has a uniformly positive scalar curvature metric of bounded geometry in the natural strict quasi-isometry class.