Aperiodic tilings, positive scalar curvature, and amenability of spaces
Aperiodic tilings, positive scalar curvature, and amenability of spaces
复制标题
非周期性平铺、正标量曲率和空间的顺应性
DOI：
10.1090/s0894-0347-1992-1145337-x
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发表时间：
1992
影响因子：
3.9
通讯作者：
S. Weinberger
中科院分区：
文献类型：
--
作者：
J. Block;S. Weinberger
本文的目的是对局部结构具有有界复杂度的非紧空间进行几何研究。这种类型的流形作为紧凑流形的叶的叶子和它们的普遍覆盖而出现。我们将使用有界复杂度的链引入一个粗糙同调理论，并研究它的一些第一性质。最有趣的结果是，当huf (X)消失时，就等周期不等式而言，它是f0ner的可适应准则的模拟和加强。(参见[4])。我们可以把这个结果看作是在任何不可调节空间上产生了一个成功的无限庞氏骗局。每个点，只有有限的资源，给它的一些邻居一些资源，但从剩余的邻居接收更多。可以想象，这对于消除非紧凑空间上的障碍物非常有用。这有很多应用。我们介绍其中的两个。第一种方法在任何不可调节的多面体上产生“不平衡”的平铺。不平衡的贴图自动是非周期性的，这就给出了许多只是非周期性贴图的贴图集的例子。不幸的是，不平衡是造成非周期性的一个特别不微妙的原因，因此我们的方法不一定能得到欧几里得空间的非周期平铺（Penrose平铺）。另一方面，使用我们的准则，大多数其他单连通非紧对称空间甚至具有不平衡平铺。第二个应用是关于泛盖具有正标量曲率的流形的特征数。我们证明了罗伊定理的一个逆定理。我们证明了在自然严格拟等距类中，对于任何不可调节群F，可以找到具有非零a属的基本群F的自旋流形，其普遍覆盖具有一致正的有界几何标量曲率度规。
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.