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Dynamics on the perfect kernel of higher rank generalized Baumslag-Solitar groups

Sasha Bontemps

TL;DR

This work generalizes the study of the space of subgroups to non-amenable GBS groups of rank $d$, describing the perfect kernel $\mathcal{K}(G)$ explicitly as the set of subgroups with infinite quotient on the Bass-Serre tree, $\{H \le G : H \backslash \mathcal{T} \text{ is infinite} \}$. It extends the rank-1 phenomena to higher rank by developing $\mathscr{H}$-preactions and $\mathscr{H}$-graphs, enabling a combinatorial encoding of subgroups and their approximations. The authors prove a countable $G$-invariant partition of $\mathcal{K}(G)$ into pieces each containing a dense $G$-orbit, establishing topological transitivity on each piece in many cases, and they identify sufficient modular-homomorphism criteria for $\mathcal{K}(G) = \mathrm{Sub}_{[\infty]}(G)$. The results unify and extend prior rank-1 and solitar– bontemps-type analyses, and they illuminate the dynamics of the conjugation action on the perfect kernel for a broad class of higher-rank GBS groups, including non-cocompact actions on the Bass–Serre tree. The paper also discusses the topology of the invariant pieces and provides explicit constructions illustrating the richness of the $G$-action on $\mathcal{K}(G)$, while highlighting cases where high transitivity does not imply dense orbits in every piece. Overall, the work advances the understanding of subgroup spaces for geometric group theory objects beyond rank one, linking Bass–Serre theory, lattice dynamics, and topological dynamics on Sub$(G)$.

Abstract

In this article, we study the space of subgroups of non-amenable generalized Baumslag-Solitar groups (GBS groups) of rank $d$, that is, groups acting cocompactly on an oriented tree with vertex and edge stabilizers isomorphic to $\mathbb{Z}^d$. Our results generalize the study of Baumslag-Solitar groups, and of GBS groups of rank $1$. We give an explicit description of the perfect kernel of a non-amenable GBS group $G$ of rank $d$ and show the existence of a partition of the perfect kernel into a countably infinite set of pieces which are invariant under the action by conjugation of $G$, and such that each piece contains a dense orbit.

Dynamics on the perfect kernel of higher rank generalized Baumslag-Solitar groups

TL;DR

This work generalizes the study of the space of subgroups to non-amenable GBS groups of rank , describing the perfect kernel explicitly as the set of subgroups with infinite quotient on the Bass-Serre tree, . It extends the rank-1 phenomena to higher rank by developing -preactions and -graphs, enabling a combinatorial encoding of subgroups and their approximations. The authors prove a countable -invariant partition of into pieces each containing a dense -orbit, establishing topological transitivity on each piece in many cases, and they identify sufficient modular-homomorphism criteria for . The results unify and extend prior rank-1 and solitar– bontemps-type analyses, and they illuminate the dynamics of the conjugation action on the perfect kernel for a broad class of higher-rank GBS groups, including non-cocompact actions on the Bass–Serre tree. The paper also discusses the topology of the invariant pieces and provides explicit constructions illustrating the richness of the -action on , while highlighting cases where high transitivity does not imply dense orbits in every piece. Overall, the work advances the understanding of subgroup spaces for geometric group theory objects beyond rank one, linking Bass–Serre theory, lattice dynamics, and topological dynamics on Sub.

Abstract

In this article, we study the space of subgroups of non-amenable generalized Baumslag-Solitar groups (GBS groups) of rank , that is, groups acting cocompactly on an oriented tree with vertex and edge stabilizers isomorphic to . Our results generalize the study of Baumslag-Solitar groups, and of GBS groups of rank . We give an explicit description of the perfect kernel of a non-amenable GBS group of rank and show the existence of a partition of the perfect kernel into a countably infinite set of pieces which are invariant under the action by conjugation of , and such that each piece contains a dense orbit.

Paper Structure

This paper contains 27 sections, 24 theorems, 86 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Graphs
  4. Space of subgroups of a countable group
  5. Graphs of groups
  6. $\mathscr{H}$-preactions and $\mathscr{H}$-graphs
  7. General setting
  8. Construction A
  9. Construction B
  10. Generalized Baumslag-Solitar groups
  11. An equivalence relation on $\mathrm{Sub}(\mathbb{Z}^d)$
  12. Case 1: Let us first assume that $|\det(\mathrm{B})| \geq 2$.
  13. Case 2 : Otherwise, $|\det(\mathrm{B})| =1$.
  14. Subcase 2.1:
  15. Subcase 2.2:
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $G$ be a non-amenable GBS group of rank $d$ defined by a reduced graph of groups $\mathscr{H}$ and let $\mathcal{T}$ be the Bass-Serre tree of $\mathscr{H}$. Then

Figures (7)

  • Figure 1: Graph of groups $\mathscr{H}$ defining an HNN-extension
  • Figure 2: Schreier graph of a $\mathscr{H}$-preaction (here $g,g' \in G_v$ and $g_e \in G_e$)
  • Figure 3: $\mathscr{H}$-graph of the $\mathscr{H}$-preaction represented in Figure \ref{['expreac']}
  • Figure 4: A graph of GBS
  • Figure 5: The graph of groups $\mathscr{H}_0$.
  • ...and 2 more figures

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Remark 2.3
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Example 3.4
  • ...and 57 more