On Finiteness of Homological Isoperimetric Functions on Top Dimensions

TL;DR

This paper proves that for groups with a finite classifying space, the homological isoperimetric function over a subfield of complex numbers is either linear or infinite, linking it to hyperbolicity in certain cases.

Contribution

It establishes a dichotomy for the homological isoperimetric function over subfields of complex numbers for groups with finite classifying spaces, extending previous results.

Findings

Homological isoperimetric function is either linear or infinite.

In groups with finite 2-dimensional classifying space, the function characterizes hyperbolicity.

The result generalizes previous work by Gersten and Mineyev.

Abstract

We address a question from \cite{BKV25} regarding the finiteness of the homological $R$-isoperimetric function. Let $R$ be a subfield of the complex numbers $\mathbb{C}$ with the absolute value norm. We prove that for any group $G$ that admits a finite $(n+1)$-dimensional model for $K(G,1)$, the homological $n$-isoperimetric function of $G$ over $R$ is either linear or takes infinite values. In particular, by results of Gersten and Mineyev, in the class of groups admitting a finite $2$-dimensional classifying space, the homological $1$-dimensional isoperimetric function over $R$ only captures hyperbolicity. This follows as a particular case of a more general result proved in this note.