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Quasi-isometry invariance of relative filling functions

Sam Hughes, Eduardo Martínez-Pedroza, Luis Jorge Sánchez Saldaña

TL;DR

This work proves that the relative Dehn function $\Delta_{G,\mathcal{P}}$ is invariant up to equivalence under quasi-isometries of pairs for relatively finitely presented pairs, provided the associated coned-off Cayley graphs are fine and the collections are reduced. It develops a framework linking fineness, almost malnormality, and coarse isoperimetric functions via coned-off graphs and induced maps on pairs, showing that hyperbolically embedded subgroups guarantee a well-defined relative Dehn function. It further connects combinatorial Dehn functions to the fineness of the 1-skeleton in cocompact simply connected $G$-2-complexes and includes Baumslag–Solitar examples illustrating the boundary of well-definedness. Overall, the results extend Osin's relative Dehn theory, offering a robust quasi-isometry invariance principle for relative isoperimetric invariants.

Abstract

For a finitely generated group $G$ and collection of subgroups $\mathcal{P}$ we prove that the relative Dehn function of a pair $(G,\mathcal{P})$ is invariant under quasi-isometry of pairs. Along the way we show quasi-isometries of pairs preserve almost malnormality of the collection and fineness of the associated coned off Cayley graphs. We also prove that for a cocompact simply connected combinatorial $G$-$2$-complex $X$ with finite edge stabilisers, the combinatorial Dehn function is well-defined if and only if the $1$-skeleton of $X$ is fine. We also show that if $H$ is a hyperbolically embedded subgroup of a finitely presented group $G$, then the relative Dehn function of the pair $(G, H)$ is well-defined. In the appendix, it is shown that show that the Baumslag-Solitar group $\mathrm{BS}(k,l)$ has a well-defined Dehn function with respect to the cyclic subgroup generated by the stable letter if and only if neither $k$ divides $l$ nor $l$ divides $k$.

Quasi-isometry invariance of relative filling functions

TL;DR

This work proves that the relative Dehn function is invariant up to equivalence under quasi-isometries of pairs for relatively finitely presented pairs, provided the associated coned-off Cayley graphs are fine and the collections are reduced. It develops a framework linking fineness, almost malnormality, and coarse isoperimetric functions via coned-off graphs and induced maps on pairs, showing that hyperbolically embedded subgroups guarantee a well-defined relative Dehn function. It further connects combinatorial Dehn functions to the fineness of the 1-skeleton in cocompact simply connected -2-complexes and includes Baumslag–Solitar examples illustrating the boundary of well-definedness. Overall, the results extend Osin's relative Dehn theory, offering a robust quasi-isometry invariance principle for relative isoperimetric invariants.

Abstract

For a finitely generated group and collection of subgroups we prove that the relative Dehn function of a pair is invariant under quasi-isometry of pairs. Along the way we show quasi-isometries of pairs preserve almost malnormality of the collection and fineness of the associated coned off Cayley graphs. We also prove that for a cocompact simply connected combinatorial --complex with finite edge stabilisers, the combinatorial Dehn function is well-defined if and only if the -skeleton of is fine. We also show that if is a hyperbolically embedded subgroup of a finitely presented group , then the relative Dehn function of the pair is well-defined. In the appendix, it is shown that show that the Baumslag-Solitar group has a well-defined Dehn function with respect to the cyclic subgroup generated by the stable letter if and only if neither divides nor divides .

Paper Structure

This paper contains 12 sections, 41 theorems, 63 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Combinatorial Dehn functions and fine graphs
  3. Coarse isoperimetric functions
  4. Relative Dehn functions of groups
  5. Fineness and quasi-isometries of pairs
  6. Almost malnormal collections and quasi-isometries of pairs
  7. Examples and non-examples
  8. Relative Dehn functions of Baumslag-Solitar groups
  9. Notation
  10. Some terminology
  11. Technical lemmas
  12. Proof of the sufficiency in Theorem \ref{['thm:BS_rel_DF-character']}

Key Result

Theorem A

Let $(G,{\mathcal{P}})\to(H, {\mathcal{Q}})$ be a quasi-isometry of pairs and let ${\mathcal{P}}^*$ be a refinement of ${\mathcal{P}}$. If the relative Dehn function $\Delta_{H,{\mathcal{Q}}}$ is well-defined, then $\Delta_{G,{\mathcal{P}}^*}$ is well-defined and $\Delta_{G,{\mathcal{P}}^\ast} \asym

Figures (3)

  • Figure 1: The image of the boundary of a ${\mathcal{P}}$-cell on $\bar{\Gamma}(G,{\mathcal{P}},S)$ under the quasi-isometry $\varphi$.
  • Figure 2: Illustration of the proof of Proposition \ref{['prop:FineQI-2']}
  • Figure 3: The cycle $q_s$ (markings inside the polygon represent the labels of the subpaths $p_1,\dots,p_8$)

Theorems & Definitions (100)

  • Theorem A
  • Theorem B: \ref{['thm.relDehn2complex']}
  • Theorem C: \ref{['thm:fine']}
  • Theorem D: \ref{['thm:malnormal']}
  • Theorem E: See \ref{['thm:ConefOff']}
  • Theorem F: Osin
  • proof : Proof of Theorem \ref{['thmx:DehnQI']}
  • Theorem G: \ref{['thm:last']}
  • Theorem 2.1: Theorem \ref{['thmx.relDehn2complex']}
  • Lemma 2.2
  • ...and 90 more