Finiteness of Homological Filling Functions

TL;DR

This paper introduces a generalized homological filling function for groups and modules, proving finiteness under certain finiteness conditions and establishing its growth as an invariant, thus addressing a key question in geometric group theory.

Contribution

It extends the concept of filling functions to modules and proves finiteness for groups of type FP_{d+1}, linking algebraic finiteness to geometric invariants.

Findings

The homological filling function is finite for modules of type FP_{d+1}.

The asymptotic growth class of the filling function is an invariant.

For groups of type FP_{d+1}, the (d+1)-dimensional filling function is finite.

Abstract

Let $G$ be a group. For any $\mathbb{Z} G$--module $M$ and any integer $d>0$, we define a function $FV_{M}^{d+1}\colon \mathbb{N} \to \mathbb{N} \cup \{\infty\}$ 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 $FP_{d+1}$ 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, addressing a question from [12].