# Variations on the theme of the uniform boundary condition

## TL;DR

This paper explores the uniform boundary condition in normed chain complexes, providing geometric F{\

## Contribution

It offers an alternative geometric proof of uniform boundary condition results, enabling integral refinements and applications to integral foliated simplicial volume.

## Key findings

- Spaces with amenable fundamental group satisfy the uniform boundary condition in all degrees.
- Geometric F{\
- paper_type":"theoretical"} } }

## Abstract

The uniform boundary condition in a normed chain complex asks for a uniform linear bound on fillings of null-homologous cycles. For the $\ell^1$-norm on the singular chain complex, Matsumoto and Morita established a characterisation of the uniform boundary condition in terms of bounded cohomology. In particular, spaces with amenable fundamental group satisfy the uniform boundary condition in every degree. We will give an alternative proof of statements of this type, using geometric F{\o}lner 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.