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Stable subgroups of graph products

Sahana H Balasubramanya, Marissa Chesser, Alice Kerr, Johanna Mangahas, Marie Trin

TL;DR

This work characterizes stable subgroups of graph products of infinite groups by linking stability to a quasi-isometric embedding into the contact graph of the prism complex, and to the algebraic notion of being almost join-free. The authors develop disk diagram and combing techniques to prove that almost join-free subgroups embed qi into the contact graph and that stable subgroups are exactly almost join-free; conversely, qi-embedded subgroups are stable. They further show that, in the torsion-free setting, stable subgroups coincide with purely loxodromic subgroups for the contact-graph action, recovering the RAAG result and extending it to general graph products. The prism complex and its contact graph provide a universal recognizing space for stability, with consequences such as the stable subgroups being virtually free. The work combines coarse geometry, van Kampen/Disk-diagram methods, and hyperbolic/cylindrical action theory to generalize the stability literature beyond RAAGs.

Abstract

We extend the characterization of stable subgroups of right-angled Artin groups of Koberda, Mangahas and Taylor to the case of graph products of infinite groups. Specifically, we show that the stable subgroups of such graph products are exactly the subgroups that quasi-isometrically embed in the associated contact graph. Equivalently, they are the subgroups that satisfy a condition arising from the defining graph: a stable subgroup is an almost join-free subgroup. In particular, we recover the equivalence between stable and purely loxodromic subgroups of Koberda, Mangahas and Taylor in the case where all torsion subgroups of the vertex groups are finite.

Stable subgroups of graph products

TL;DR

This work characterizes stable subgroups of graph products of infinite groups by linking stability to a quasi-isometric embedding into the contact graph of the prism complex, and to the algebraic notion of being almost join-free. The authors develop disk diagram and combing techniques to prove that almost join-free subgroups embed qi into the contact graph and that stable subgroups are exactly almost join-free; conversely, qi-embedded subgroups are stable. They further show that, in the torsion-free setting, stable subgroups coincide with purely loxodromic subgroups for the contact-graph action, recovering the RAAG result and extending it to general graph products. The prism complex and its contact graph provide a universal recognizing space for stability, with consequences such as the stable subgroups being virtually free. The work combines coarse geometry, van Kampen/Disk-diagram methods, and hyperbolic/cylindrical action theory to generalize the stability literature beyond RAAGs.

Abstract

We extend the characterization of stable subgroups of right-angled Artin groups of Koberda, Mangahas and Taylor to the case of graph products of infinite groups. Specifically, we show that the stable subgroups of such graph products are exactly the subgroups that quasi-isometrically embed in the associated contact graph. Equivalently, they are the subgroups that satisfy a condition arising from the defining graph: a stable subgroup is an almost join-free subgroup. In particular, we recover the equivalence between stable and purely loxodromic subgroups of Koberda, Mangahas and Taylor in the case where all torsion subgroups of the vertex groups are finite.

Paper Structure

This paper contains 22 sections, 32 theorems, 18 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coarse geometry
  4. Stable subgroups
  5. Graph products
  6. Join, star, and link subgroups
  7. Associated graphs and complexes
  8. Prism Cayley graph and complex
  9. Hyperplanes and carriers
  10. Contact graph
  11. Star Cayley graph
  12. Disk diagrams
  13. Constructing disk diagrams
  14. Disk diagram lemmas
  15. Combing
  16. ...and 7 more sections

Key Result

Theorem 1.1

RAAGstable Let $\Gamma$ be a finite connected graph and $H$ a finitely generated subgroup of the right angled Artin group $A(\Gamma)$. The following are equivalent:

Figures (13)

  • Figure 1: 2--cells of the prism complex
  • Figure 2: Midcubes for 1-- and 2--cells of $P(\Gamma\mathcal{G})$
  • Figure 3: Hyperplanes, geometric hyperplanes, and contact graph. $H_1$ and $H_2$, $H_2$ and $H_3$, and $H_3$ and $H_4$ are transverse, while $H_1$ and $H_3$, and $H_2$ and $H_4$ are tangent.
  • Figure 4: From van Kampen diagram to dual van Kampen diagram
  • Figure 5: From (dual) van Kampen diagram to disk diagram
  • ...and 8 more figures

Theorems & Definitions (71)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.5
  • Remark 1.7
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • ...and 61 more