Isoperimetric inequalities in finitely generated groups

TL;DR

This paper introduces the Dehn spectrum, a new quasi-isometric invariant for finitely generated groups, extending isoperimetric analysis beyond finitely presented groups and exploring its structure and implications.

Contribution

It defines and studies the Dehn spectrum for finitely generated groups, computes it for specific classes, and investigates the diversity of groups with finite exponent.

Findings

Dehn spectrum encodes isoperimetric behavior at various scales.

Computed Dehn spectrum for small cancellation groups, wreath products, and free Burnside groups.

Proved existence of continuum many non-quasi-isometric finitely generated groups of finite exponent.

Abstract

To each finitely generated group $G$, we associate a quasi-isometric invariant called the \emph{Dehn spectrum} of $G$. If $G$ is finitely presented, our invariant is closely related to the Dehn function of $G$, but provides more information by encoding the isoperimetric behavior of $G$ at various scales. The main goal of this paper is to initiate the study of the Dehn spectrum of finitely generated (but not necessarily finitely presented) groups. In particular, we compute the Dehn spectrum of small cancellation groups, certain wreath products, and free Burnside groups of sufficiently large odd exponent. We also address several natural questions concerning the structure of the poset of Dehn spectra. As an application, we show that there exist $2^{\aleph_0}$ pairwise non-quasi-isometric finitely generated groups of finite exponent.