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Perfect kernel of generalized Baumslag-Solitar groups

Sasha Bontemps

Abstract

In this article, we study the space of subgroups of generalized Baumslag-Solitar groups (GBS groups), that is, groups acting cocompactly on an oriented tree without inversion and with infinite cyclic vertex and edge stabilizers. Our results generalize the study of Baumslag-Solitar groups in [CGLMS22]. Given a GBS group G defined by a graph of groups whose existence is given by Bass-Serre theory, we associate to any subgroup of G an integer, which is a generalization of the phenotype defined in [CGLMS22]. This quantity is invariant under conjugation and allows us to decompose the perfect kernel of G into pieces which are invariant under conjugation and on which G acts highly topologically transitively. To achieve this, we interpret graphs of subgroups of G as "blown up and shrunk" Schreier graphs of transitive actions of G. We also describe the topology of the pieces which appear in the decomposition.

Perfect kernel of generalized Baumslag-Solitar groups

Abstract

In this article, we study the space of subgroups of generalized Baumslag-Solitar groups (GBS groups), that is, groups acting cocompactly on an oriented tree without inversion and with infinite cyclic vertex and edge stabilizers. Our results generalize the study of Baumslag-Solitar groups in [CGLMS22]. Given a GBS group G defined by a graph of groups whose existence is given by Bass-Serre theory, we associate to any subgroup of G an integer, which is a generalization of the phenotype defined in [CGLMS22]. This quantity is invariant under conjugation and allows us to decompose the perfect kernel of G into pieces which are invariant under conjugation and on which G acts highly topologically transitively. To achieve this, we interpret graphs of subgroups of G as "blown up and shrunk" Schreier graphs of transitive actions of G. We also describe the topology of the pieces which appear in the decomposition.

Paper Structure

This paper contains 23 sections, 28 theorems, 120 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Graphs
  4. Definitions and notations
  5. Schreier graphs
  6. Elements of Bass-Serre theory
  7. Generalized Baumslag-Solitar groups
  8. Space of subgroups and perfect kernel
  9. H-preactions and H-graphs
  10. From an H-preaction to an H-graph
  11. From an H-graph to an H-preaction
  12. Phenotype of a GBS group
  13. Some arithmetical properties of the H-phenotype
  14. Base case:
  15. Induction step
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\Gamma$ be a non amenable GBS group. Then

Figures (6)

  • Figure 1: The graph $\mathcal{H}$.
  • Figure 2: A collection of partial bijections which cannot be extended into a genuine $\Gamma$-action.
  • Figure 3: Construction A.
  • Figure 4: Construction B.
  • Figure 10: Base case (case of a segment).
  • ...and 1 more figures

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Definition 2.1
  • Proposition 2.1
  • proof
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.1
  • ...and 51 more