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 $F{V}_M^{d+1}:\mathbb{N}\to \mathbb{N}\cup \{\infty \}$ 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 $F{P}_{d+1}$ 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 $(d+1)$-dimensional homological filling function takes only finite values, addressing a question from [12].

Minor typo in the statement of Theorem 1.3 was corrected

References in corpus (1)

• A note on fine graphs and homological isoperimetric inequalities

Cited by in corpus (1)

• Subgroups, hyperbolicity and cohomological dimension for totally disconnected locally compact groups