Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Subgroup mixing in Baumslag-Solitar groups

Sasha Bontemps

TL;DR

This work investigates the dynamics of conjugation on the space of subgroups of the Baumslag-Solitar groups $BS(m,n)$ via random walks. By organizing the perfect kernel into the infinite phenotype partition $Ph_{m,n}$ with associated sets $\mathcal{K}_N$, the authors isolate when the conjugation action is topologically $\mu$-mixing: nonunimodular, non-metabelian cases exhibit non-mixing on finite-phenotype pieces, while the unimodular case yields $\mu$-mixing on the infinite piece $\mathcal{K}_{\infty}$ and on all finite pieces when $|m|=|n|$. The proofs combine Bass-Serre theory, $(m,n)$-graphs (Schreier graphs) with refined random-walk arguments, including a novel transfer-inequality analysis and a maximal forest saturation framework. Overall, the paper extends mixing phenomena from acylindrically hyperbolic settings to a non-acylindrical, non-unimodular class of Baumslag-Solitar groups, clarifying the intricate dynamics on the space of subgroups under conjugation.

Abstract

In this article, we contribute to the study of the dynamics induced by the conjugation action on the space of subgroups of Baumslag-Solitar groups BS(m,n), via the mixing properties of elements asymptotically produced by suitable random walks on the group. In an acylindrically hyperbolic context, the authors of [HMO] demonstrated strong mixing situations, namely topological mu-mixing, a strengthening of high topological transitivity. Regarding non-metabelian and non-unimodular BS(m,n), we exhibit here a radically different situation on each of the pieces except one of the partition introduced in [CGLMS22] (although it is highly topologically transitive on each piece). On the other hand, when BS(m,n) is unimodular, we demonstrate the topological mu-mixing character on each of the pieces.

Subgroup mixing in Baumslag-Solitar groups

TL;DR

This work investigates the dynamics of conjugation on the space of subgroups of the Baumslag-Solitar groups via random walks. By organizing the perfect kernel into the infinite phenotype partition with associated sets , the authors isolate when the conjugation action is topologically -mixing: nonunimodular, non-metabelian cases exhibit non-mixing on finite-phenotype pieces, while the unimodular case yields -mixing on the infinite piece and on all finite pieces when . The proofs combine Bass-Serre theory, -graphs (Schreier graphs) with refined random-walk arguments, including a novel transfer-inequality analysis and a maximal forest saturation framework. Overall, the paper extends mixing phenomena from acylindrically hyperbolic settings to a non-acylindrical, non-unimodular class of Baumslag-Solitar groups, clarifying the intricate dynamics on the space of subgroups under conjugation.

Abstract

In this article, we contribute to the study of the dynamics induced by the conjugation action on the space of subgroups of Baumslag-Solitar groups BS(m,n), via the mixing properties of elements asymptotically produced by suitable random walks on the group. In an acylindrically hyperbolic context, the authors of [HMO] demonstrated strong mixing situations, namely topological mu-mixing, a strengthening of high topological transitivity. Regarding non-metabelian and non-unimodular BS(m,n), we exhibit here a radically different situation on each of the pieces except one of the partition introduced in [CGLMS22] (although it is highly topologically transitive on each piece). On the other hand, when BS(m,n) is unimodular, we demonstrate the topological mu-mixing character on each of the pieces.
Paper Structure (17 sections, 14 theorems, 77 equations, 1 figure)

This paper contains 17 sections, 14 theorems, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Space of subgroups
  4. Baumslag-Solitar groups; preactions and $(m,n)$-(Schreier) graphs
  5. Some background on Bass-Serre theory
  6. Normal forms
  7. Preactions and $(m,n)$-graphs
  8. Random walks on groups
  9. Phenotype of a subgroup of $\mathrm{BS}(m,n)$
  10. Random walks on the Bass-Serre tree and on $(m,n)$-graphs
  11. Proof of Theorem \ref{['nonmixing']}
  12. Base case
  13. Induction step
  14. 1st case: $g_{i+1} \in \{b,b^{-1}\}$.
  15. 2nd case: $g_{i+1} = t$.
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $m,n \in \mathbb{Z}$ such that $\min(|m|,|n|) > 1$. Let us assume that $|m| \neq |n|$. Then, there exists a probability measure $\mu$ whose support is finite, symmetric and generates $\mathrm{BS}(m,n)$, such that for every finite $P \in \mathcal{Q}_{m,n}$, the action by conjugation of $\mathrm{B

Figures (1)

  • Figure 1: An illustration of the proof of Lemma \ref{['merge']}

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Example 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4: Maximal forest saturation
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • ...and 22 more