On Finiteness of Homological Isoperimetric Functions on Top Dimensions

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.