Bounded degree cosystolic expanders of every dimension
Bounded degree cosystolic expanders of every dimension
复制标题
每个维度的有界度共收缩扩张器
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
T. Kaufman
中科院分区：
文献类型：
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作者：
Shai Evra;T. Kaufman
近年来，出现了高维的扩张器理论。图的组合膨胀的概念（即图的cheeger常数）已经看到了对高维简单复合物的两个概括。一种被称为串联扩张的概括是由于海洋和慕尼黑造成的。我们在这里称其为宇宙扩张的另一个是由于Gromov造成的，Gromov表明宇宙扩张器具有拓扑重叠的特性。直到Kaufman，Kazhdan和Lubotzky的最新工作，该构造程度组合扩展器（根据任何一个定义）都没有构造（随机或显式），该作品提供了第二个维度二维的界面扩张器。在较高的维度中，无界度组合扩展器是已知的。在这项工作中，我们介绍了每个维度的明确界限宇宙扩张器。这解决了格罗莫夫（Gromov）提出的一个悬而未决的问题，他询问在每个维度中是否存在与拓扑重叠属性有限的综合体。此外，我们在一个复杂的复合物上提供了一个局部到全球标准，这意味着宇宙的扩展：即，对于D维复合体而言，如果其基础图是一个很好的扩展器，并且其所有链接既是均匀的扩张器又是良好的扩展器图，那么然后，该络合物的（D-1）维骨架是宇宙扩张器。
In recent years a high dimensional theory of expanders has emerged. The notion of combinatorial expansion of graphs (i.e. the Cheeger constant of a graph) has seen two generalizations to high dimensional simplicial complexes. One generalization, known as coboundary expansion, is due to Linial and Meshulem; the other, which we term here cosystolic expansion, is due to Gromov, who showed that cosystolic expanders have the topological overlapping property. No construction (either random or explicit) of bounded degree combinational expanders (according to either definition) were known until a recent work of Kaufman, Kazhdan and Lubotzky, which provided the first bounded degree cosystolic expanders of dimension two. No bounded degree combinatorial expanders are known in higher dimensions. In this work we present explicit bounded degree cosystolic expanders of every dimension. This solves affirmatively an open question raised by Gromov, who asked whether there exist bounded degree complexes with the topological overlapping property in every dimension. Moreover, we provide a local to global criterion on a complex that implies cosystolic expansion: Namely, for a d-dimensional complex, X, if its underlying graph is a good expander, and all its links are both coboundary expanders and good expander graphs, then the (d-1)-dimensional skeleton of the complex is a cosystolic expander.