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Coarse separation and splittings in right-angled Artin groups

Oussama Bensaid, Anthony Genevois, Romain Tessera

Abstract

In this article, we characterise geometrically when a right-angled Artin group splits over an abelian subgroup. More precisely, given a finite graph $Γ$, we show that $A(Γ)$ splits over an abelian subgroup if and only if it is coarsely separable by a family of subexponential growth, which amounts to saying that $Γ$ is complete or separated by a complete subgraph.

Coarse separation and splittings in right-angled Artin groups

Abstract

In this article, we characterise geometrically when a right-angled Artin group splits over an abelian subgroup. More precisely, given a finite graph , we show that splits over an abelian subgroup if and only if it is coarsely separable by a family of subexponential growth, which amounts to saying that is complete or separated by a complete subgraph.

Paper Structure

This paper contains 25 sections, 45 theorems, 54 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Discussion.
  3. Preliminaries
  4. Coarse separation in acylindrically hyperbolic groups
  5. Morse geodesics
  6. Separating Morse geodesics
  7. Coarse separation in thick spaces
  8. Coarse separation in graph products
  9. Quasi-median geometry
  10. Quasi-median graphs.
  11. Systems of local metrics.
  12. Graph products.
  13. Invasive subgraphs
  14. Proof of Theorem \ref{['thm:CoarseSepGP']}
  15. Coarse separation in right-angled Artin groups
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\Gamma$ be a finite graph. The following assertions are equivalent:

Figures (10)

  • Figure 1: A few hyperplanes in a quasi-median graph.
  • Figure 2: Examples satisfying $(1)$, $(2)$, or both.
  • Figure 3: Counterexamples in the presence of triangles.
  • Figure 4: Case 1.
  • Figure 5: Case 2.
  • ...and 5 more figures

Theorems & Definitions (101)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Proposition 1.6
  • Conjecture 1.7
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • ...and 91 more