Finiteness of homological filling functions

Abstract

Let $G$ be a group. For any $\mathbb{Z}G$-module $M$ and any integer $d>0$, we define a function ${FV}_M^{d+1}:\mathbb{N}\to \mathbb{N}\cup \left\{\infty \right\}$ generalizing the notion of $\left(d+1\right)$-dimensional filling function of a group. We prove that this function takes only finite values if $M$ is of type $F{P}_{d+1}$ 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 $F{P}_{d+1}$, our main result implies that its $\left(d+1\right)$-dimensional homological filling function takes only finite values.