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Isoperimetric inequalities in finitely generated groups

D. Osin, E. Rybak

TL;DR

This work introduces the Dehn spectrum $f_G$ as a three-variable, quasi-isometric invariant for finitely generated groups, refining the classical Dehn function by capturing isoperimetric behavior across scales. It establishes invariance under quasi-isometries, relates the spectrum to the Dehn function for finitely presented groups, and develops a descriptive-set-theoretic approach to spectra over marked groups. The authors compute linear Dehn spectra for several natural families (small cancellation groups, wreath products, and large-odd-exponent Burnside groups) and show, via descriptive and geometric methods, that there are continuum many non-quasi-isometric finitely generated groups of finite exponent, as well as continuum many spectra. These results illuminate the landscape of isoperimetric behavior at multiple scales and demonstrate the spectrum’s richness beyond the classical Dehn function. The paper concludes with open questions on realizability, broader computations, and the interplay between spectra and large-scale geometry, including asymptotic cones and central extensions.

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.

Isoperimetric inequalities in finitely generated groups

TL;DR

This work introduces the Dehn spectrum as a three-variable, quasi-isometric invariant for finitely generated groups, refining the classical Dehn function by capturing isoperimetric behavior across scales. It establishes invariance under quasi-isometries, relates the spectrum to the Dehn function for finitely presented groups, and develops a descriptive-set-theoretic approach to spectra over marked groups. The authors compute linear Dehn spectra for several natural families (small cancellation groups, wreath products, and large-odd-exponent Burnside groups) and show, via descriptive and geometric methods, that there are continuum many non-quasi-isometric finitely generated groups of finite exponent, as well as continuum many spectra. These results illuminate the landscape of isoperimetric behavior at multiple scales and demonstrate the spectrum’s richness beyond the classical Dehn function. The paper concludes with open questions on realizability, broader computations, and the interplay between spectra and large-scale geometry, including asymptotic cones and central extensions.

Abstract

To each finitely generated group , we associate a quasi-isometric invariant called the \emph{Dehn spectrum} of . If is finitely presented, our invariant is closely related to the Dehn function of , but provides more information by encoding the isoperimetric behavior of 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 pairwise non-quasi-isometric finitely generated groups of finite exponent.
Paper Structure (19 sections, 62 theorems, 114 equations, 13 figures)

This paper contains 19 sections, 62 theorems, 114 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Basic properties of Dehn spectrum
  4. Notation.
  5. Dehn spectrum of a graph
  6. Quasi-isometric invariance
  7. The case of finitely presented groups
  8. Groups with linear Dehn spectrum
  9. Limits of hyperbolic groups
  10. Wreath products
  11. Geodesics and detours in Cayley graphs
  12. A descriptive approach to isoperimetric functions
  13. $F_\sigma$ relations on Polish spaces
  14. The Dehn spectrum of marked groups
  15. Groups of finite exponent
  16. ...and 4 more sections

Key Result

Theorem 1.4

Let $G$ and $H$ be groups generated by finite sets $X$ and $Y$, respectively. If $G$ and $H$ are quasi-isometric, then $f_{G,X} \sim f_{H,Y}$. In particular, the equivalence class of the Dehn spectrum of a finitely generated group is independent of the choice of a particular generating set.

Figures (13)

  • Figure 1: Constructing a combinatorial filling of $c$.
  • Figure 2: Case 2 in the proof of the upper bound in Theorem \ref{['Thm:FP']}
  • Figure 3: Modifying $\widetilde{D}$
  • Figure 4: The map $\psi\colon \mathop{\mathrm{Cay}}\nolimits(G,X) \to T$.
  • Figure 5: An $r$-detour.
  • ...and 8 more figures

Theorems & Definitions (134)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Definition 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 124 more