The uniform boundary condition (UBC) in a normed chain complex asks for a uniform linear bound on fillings of null-homologous cycles. For the ℓ1-norm on the singular chain complex, Matsumoto and Morita established a characterization of the UBC in terms of bounded cohomology. In particular, spaces with amenable fundamental group satisfy the UBC in every degree. We will give an alternative proof of statements of this type, using geometric Følner arguments on the chain level instead of passing to the dual cochain complex. These geometric methods have the advantage that they also lead to integral refinements. In particular, we obtain applications in the context of integral foliated simplicial volume.

中文翻译：

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统一边界条件主题的变体

范数链复形中的统一边界条件 (UBC) 要求对零同源循环的填充有统一的线性边界。为了ℓ1-范数关于奇异链复形，松本和森田根据有界上同调建立了 UBC 的表征。特别是，具有服从基本群的空间在各个方面都满足 UBC。我们将给出这种类型陈述的替代证明，在链级别使用几何 Følner 参数，而不是传递给对偶 cochain 复合体。这些几何方法的优点是它们也导致积分细化。特别是，我们在积分叶形单纯体积的背景下获得了应用。