Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Coarse separation and splittings in hyperbolic groups

Oussama Bensaid, Anthony Genevois, Romain Tessera

Abstract

We study coarse separation in one-ended hyperbolic groups from a quantitative point of view, focusing on the volume growth of separating subsets. We prove that a one-ended hyperbolic group that is not virtually a surface group is coarsely separable by a subset of subexponential growth if and only if it splits over a virtually cyclic subgroup. To do so, we show that sufficiently large thickened spheres are hard to cut, in the sense that their cut-sets have exponential size, a result of independent interest. As an application, we obtain a polynomial lower bound on the separation profile of one-ended hyperbolic groups that do not split over a two-ended subgroup. We also apply our criterion to graph products of finite groups, giving a combinatorial characterisation of when such graph products are coarsely separable by a subset of subexponential growth.

Coarse separation and splittings in hyperbolic groups

Abstract

We study coarse separation in one-ended hyperbolic groups from a quantitative point of view, focusing on the volume growth of separating subsets. We prove that a one-ended hyperbolic group that is not virtually a surface group is coarsely separable by a subset of subexponential growth if and only if it splits over a virtually cyclic subgroup. To do so, we show that sufficiently large thickened spheres are hard to cut, in the sense that their cut-sets have exponential size, a result of independent interest. As an application, we obtain a polynomial lower bound on the separation profile of one-ended hyperbolic groups that do not split over a two-ended subgroup. We also apply our criterion to graph products of finite groups, giving a combinatorial characterisation of when such graph products are coarsely separable by a subset of subexponential growth.
Paper Structure (18 sections, 28 theorems, 86 equations, 1 figure)

This paper contains 18 sections, 28 theorems, 86 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Coarse separation and splittings.
  3. Strategy.
  4. Applications of our techniques.
  5. Preliminaries
  6. Coarse separation and cuts
  7. Hyperbolic groups and their boundaries
  8. Gromov boundary
  9. Uniform distortion of spheres in one-ended hyperbolic groups
  10. Shadows
  11. Proof of the main theorem
  12. Applications to graph products
  13. Graph products
  14. Hyperbolicity
  15. When virtually cyclic.
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $G$ be a hyperbolic group. Assume that $G$ is one-ended and not virtually a surface group. Then $G$ is coarsely separable by a family of subexponential growth if and only if it splits over a virtually cyclic subgroup.

Figures (1)

  • Figure 1: The two graph products from Example \ref{['ex:CoarseEmb']}.

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • ...and 50 more