#### Vol. 11, No. 4, 2018

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Finiteness of homological filling functions

### Joshua W. Fleming and Eduardo Martínez-Pedroza

Vol. 11 (2018), No. 4, 569–583
##### Abstract

Let $G$ be a group. For any $ℤG$-module $M$ and any integer $d>0$, we define a function ${FV}_{M}^{d+1}:ℕ\to ℕ\cup \left\{\infty \right\}$ 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.

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##### Keywords
Dehn functions, homological filling function, isoperimetric inequalities, finiteness properties of groups
##### Mathematical Subject Classification 2010
Primary: 20F65, 20J05
Secondary: 16P99, 28A75, 57M07