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Explicit polynomial bounds on Dehn functions of subgroups of hyperbolic groups

Robert Kropholler, Claudio Llosa Isenrich, Ignat Soroko

TL;DR

This work provides the first explicit polynomial upper bound on the Dehn function of a non-hyperbolic finitely presented subgroup of a hyperbolic group, focusing on Brady’s example. It develops a geometric approach based on CAT(0) cube complexes and a height function, pushing fillings into level sets and combining δ-thin triangle controls with ascending/descending link data to yield a bound of $n^{96}$ for the Dehn function, with a quadratic lower bound $n^2$. A key auxiliary result is the precise computation that the 1-skeleton of Brady’s cube complex is $4$-point hyperbolic, which underpins the exponent in the Dehn bound. The methods provide a framework for computing explicit Dehn-function bounds in similar subgroups and offer insights into the geometry of non-hyperbolic subgroups within hyperbolic groups.

Abstract

In 1999 Brady constructed the first example of a non-hyperbolic finitely presented subgroup of a hyperbolic group by fibring a non-positively curved cube complex over the circle. We show that his example has Dehn function bounded above by $n^{96}$. This provides the first explicit polynomial upper bound on the Dehn function of a finitely presented non-hyperbolic subgroup of a hyperbolic group. We also determine the precise hyperbolicity constant for the $1$-skeleton of the universal cover of the cube complex in Brady's construction with respect to the $4$-point condition for hyperbolicity.

Explicit polynomial bounds on Dehn functions of subgroups of hyperbolic groups

TL;DR

This work provides the first explicit polynomial upper bound on the Dehn function of a non-hyperbolic finitely presented subgroup of a hyperbolic group, focusing on Brady’s example. It develops a geometric approach based on CAT(0) cube complexes and a height function, pushing fillings into level sets and combining δ-thin triangle controls with ascending/descending link data to yield a bound of for the Dehn function, with a quadratic lower bound . A key auxiliary result is the precise computation that the 1-skeleton of Brady’s cube complex is -point hyperbolic, which underpins the exponent in the Dehn bound. The methods provide a framework for computing explicit Dehn-function bounds in similar subgroups and offer insights into the geometry of non-hyperbolic subgroups within hyperbolic groups.

Abstract

In 1999 Brady constructed the first example of a non-hyperbolic finitely presented subgroup of a hyperbolic group by fibring a non-positively curved cube complex over the circle. We show that his example has Dehn function bounded above by . This provides the first explicit polynomial upper bound on the Dehn function of a finitely presented non-hyperbolic subgroup of a hyperbolic group. We also determine the precise hyperbolicity constant for the -skeleton of the universal cover of the cube complex in Brady's construction with respect to the -point condition for hyperbolicity.
Paper Structure (18 sections, 22 theorems, 12 equations, 19 figures)

This paper contains 18 sections, 22 theorems, 12 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Hyperbolic groups and their non-hyperbolic subgroups
  3. Dehn functions of subgroups of hyperbolic groups
  4. Idea of proof of
  5. Structure
  6. Acknowledgements
  7. Background
  8. Dehn functions
  9. Hyperbolic spaces and groups
  10. CAT(0) cube complexes and Morse theory
  11. Brady's example
  12. A geometric version of a Theorem of Gersten and Short
  13. A theorem of Gersten and Short
  14. Fillings in cube complexes
  15. Pushing fillings into level sets
  16. ...and 3 more sections

Key Result

Theorem 1.1

There is a hyperbolic group $G$ with a finitely presented subgroup $H\leq G$ which has Dehn function $\delta_H$ satisfying $n^2\preccurlyeq \delta_H(n)\preccurlyeq n^{96}$.

Figures (19)

  • Figure 1: The filling of a loop $\gamma$ in the proof of Lemma \ref{['lem:hyperbolic-area-radius-pairs']}
  • Figure 2: Pushing the $1$-skeleton around a vertex $v\in C$.
  • Figure 3: Pushing a $2$-cell $\Delta$ in $C$ into the $(m-k-1)$-level set
  • Figure 4: The sliced cell structure on a $3$-cube with coloured top and bottom triangles
  • Figure 5: A connected neighbourhood of a boundary vertex inside $N_1(C)\setminus C$
  • ...and 14 more figures

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.2: Alonso Alo-90
  • Corollary 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • Lemma 2.7
  • proof
  • ...and 38 more