The Cheeger constant is a measure of the edge expansion of a graph and as such plays a key role in combinatorics and theoretical computer science. In recent years, there is an interest in k-dimensional versions of the Cheeger constant that likewise provide quantitative measure of cohomological acyclicity of a complex in dimension k. In this paper, we study several aspects of the higher Cheeger constants. Our results include methods for bounding the cosystolic norm of k-cochains and the k-th Cheeger constants, with applications to the expansion of pseudomanifolds, Coxeter complexes and homogenous geometric lattices. We revisit a theorem of Gromov on the expansion of a product of a complex with a simplex and provide an elementary derivation of the expansion in a hypercube. We prove a lower bound on the maximal cosystole in a complex and an upper bound on the expansion of bounded degree complexes and give an essentially sharp estimate for the cosystolic norm of the Paley cochains. Finally, we discuss a non-abelian version of the one-dimensional expansion of a simplex, with an application to a question of Babson on bounded quotients of the fundamental group of a random 2-complex.

中文翻译：

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非周期性的定量方面

Cheeger常数是图形边缘扩展的量度，因此在组合和理论计算机科学中起着关键作用。近年来，有在关心ķ的Cheeger常数，同样提供维复杂的上同调acyclicity的定量测量的三维版本ķ。在本文中，我们研究了更高的Cheeger常数的几个方面。我们的结果包括限制k -cochains和k的cosystolic范数的方法Cheeger常数，可应用于拟流形，Coxeter络合物和均匀几何格的展开。我们重新讨论关于具有单形的复杂产品的展开的Gromov定理，并提供超立方体中展开的基本推导。我们证明了复合物中最大cosystole的下限和有界度复合物扩展的上限，并给出了对Paley cochains的cosystolic范本的基本清晰估计。最后，我们讨论了单纯形的一维展开的非阿贝尔版本，并将其应用于关于随机2复数基本群的有界商的Babson问题。