In this paper we provide a framework for the study of isoperimetric problems in finitely generated groups, through a combinatorial study of universal covers of compact simplicial complexes. We show that, when estimating filling functions, one can restrict to simplicial spheres of particular shapes, called “round” and “unfolded”, provided that a bounded quasi-geodesic combing exists. We prove that the problem of estimating higher dimensional divergence as well can be restricted to round spheres. Applications of these results include a combinatorial analogy of the Federer–Fleming inequality for finitely generated groups.

中文翻译：

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组合高维等距和散度

在本文中，我们通过对紧单纯复形的普遍覆盖的组合研究，为研究有限生成群中的等周问题提供了一个框架。我们表明，在估计填充函数时，只要存在有界准测地线组合，就可以限制为特定形状的单纯球体，称为“圆形”和“展开”。我们证明了估计高维散度的问题也可以限制在圆形球体上。这些结果的应用包括对有限生成群的费德勒-弗莱明不等式的组合类比。