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On Finiteness of Homological Isoperimetric Functions on Top Dimensions

Eduardo Martínez-Pedroza, Diana Vizcaíno Torres

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.

On Finiteness of Homological Isoperimetric Functions on Top Dimensions

Abstract

We address a question from \cite{BKV25} regarding the finiteness of the homological -isoperimetric function. Let be a subfield of the complex numbers with the absolute value norm. We prove that for any group that admits a finite -dimensional model for , the homological -isoperimetric function of over is either linear or takes infinite values. In particular, by results of Gersten and Mineyev, in the class of groups admitting a finite -dimensional classifying space, the homological -dimensional isoperimetric function over only captures hyperbolicity. This follows as a particular case of a more general result proved in this note.
Paper Structure (2 sections, 6 theorems, 21 equations)

This paper contains 2 sections, 6 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Proof of the theorem

Key Result

theorem 1

Let $R$ be a normed division ring that respects the rationals. Let $G$ be an $(n+1)$-dimensional admissible group over $R$. Then the homological isoperimetric function $f_n$ with respect to $R$ takes only finite values if and only if $f_n$ is bounded by a linear function.

Theorems & Definitions (12)

  • definition 1
  • theorem 1
  • corollary 1
  • corollary 2
  • remark 1
  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • ...and 2 more