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The uniform boundary condition and simplicial volumes

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NPCW05 - Group actions and cohomology in non-positive curvature

Co-author: Daniel Fauser (Universität Regensburg)

The uniform boundary condition on a normed chain complex requires the existence of controlled fillings for all boundaries. The uniform boundary condition naturally comes up in the context of glueing results for simplicial volume. Matsumoto and Morita showed that the singular chain complex of spaces with amenable fundamental group satisfies the uniform boundary condition, using bounded cohomology. We give a direct geometric proof of this fact in the aspherical case. This proof admits generalisations to integral foliated simplicial volume, which provides upper bounds for \\ -Betti numbers and the rank gradient.

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