Finiteness of homological filling functions. (English) Zbl 1494.20060

Summary: Let \(G\) be a group. For any \(\mathbb{Z}G\)-module \(M\) and any integer \(d>0\), we define a function \(\operatorname{FV}_{M}^{d+1}: \mathbb{N} \rightarrow \mathbb{N}\cup \{\infty\}\) generalizing the notion of \((d+1)\)-dimensional filling function of a group. We prove that this function takes only finite values if \(M\) is of type \(FP_{d+1}\) and \(d>0\), and remark that the asymptotic growth class of this function is an invariant of \(M\). In the particular case that \(G\) is a group of type \(FP_{d+1}\), our main result implies that its \((d+1)\)-dimensional homological filling function takes only finite values.


20F65 Geometric group theory
20J05 Homological methods in group theory
57M07 Topological methods in group theory
Full Text: DOI arXiv


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