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Fixed subgroups of generalised Baumslag-Solitar groups

Oli Jones, Alan Logan

TL;DR

The paper develops a comprehensive framework for fixed subgroups of automorphisms of generalised Baumslag–Solitar (GBS) groups, focusing on automorphisms that preserve a Bass–Serre tree under 1-free conditions. It establishes precise criteria for when Fix($\phi$) is finitely generated, linking this to invariants $\beta(G)$ and $\Delta(G)$, and demonstrates both finite-generation and non-finite-generation phenomena, including an infinite family of non-finitely generated examples. The work also proves that finite-order automorphisms always have finitely generated fixed subgroups and shows that if the GBS graph is a tree, all automorphisms have finitely generated fixed subgroups, using Bass–Serre theory and BNS invariants. Collectively, the results yield a near-complete classification of finite-generation and boundedness of fixed subgroups in 1-free GBS groups, and provide structural insights for tree GBS groups via deformation arguments and invariant theory.

Abstract

We investigate fixed subgroups of automorphisms of generalised Baumslag-Solitar (GBS) groups. Our main results are for automorphisms leaving a Bass-Serre tree invariant, under the assumption that all edge stabilisers are strictly contained in the corresponding vertex stabilisers. We completely characterise which GBS groups admit such an automorphism with a fixed subgroup which is not finitely-generated. In doing so, we provide an infinite family of examples of non-finitely generated fixed subgroups in GBS groups. Dropping the above assumptions, we show that all finite order automorphisms of GBS groups have finitely generated fixed subgroups. Furthermore, we show that when the GBS graph is a tree, all automorphisms have finitely generated fixed subgroups.

Fixed subgroups of generalised Baumslag-Solitar groups

TL;DR

The paper develops a comprehensive framework for fixed subgroups of automorphisms of generalised Baumslag–Solitar (GBS) groups, focusing on automorphisms that preserve a Bass–Serre tree under 1-free conditions. It establishes precise criteria for when Fix() is finitely generated, linking this to invariants and , and demonstrates both finite-generation and non-finite-generation phenomena, including an infinite family of non-finitely generated examples. The work also proves that finite-order automorphisms always have finitely generated fixed subgroups and shows that if the GBS graph is a tree, all automorphisms have finitely generated fixed subgroups, using Bass–Serre theory and BNS invariants. Collectively, the results yield a near-complete classification of finite-generation and boundedness of fixed subgroups in 1-free GBS groups, and provide structural insights for tree GBS groups via deformation arguments and invariant theory.

Abstract

We investigate fixed subgroups of automorphisms of generalised Baumslag-Solitar (GBS) groups. Our main results are for automorphisms leaving a Bass-Serre tree invariant, under the assumption that all edge stabilisers are strictly contained in the corresponding vertex stabilisers. We completely characterise which GBS groups admit such an automorphism with a fixed subgroup which is not finitely-generated. In doing so, we provide an infinite family of examples of non-finitely generated fixed subgroups in GBS groups. Dropping the above assumptions, we show that all finite order automorphisms of GBS groups have finitely generated fixed subgroups. Furthermore, we show that when the GBS graph is a tree, all automorphisms have finitely generated fixed subgroups.
Paper Structure (12 sections, 35 theorems, 21 equations, 1 figure)

This paper contains 12 sections, 35 theorems, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries: GBS groups and Compatibility
  3. Bass-Serre Theory
  4. Generalised Baumslag--Solitar (GBS) groups
  5. Presentations and Fundamental Domains
  6. Automorphisms and compatible actions
  7. Fixed subgroups of compatible automorphisms
  8. Automorphisms not fixing a point of $T$
  9. Automorphisms with sign $-1$
  10. Automorphisms with sign $1$
  11. Non-finitely generated fixed subgroups
  12. Fixed subgroups of tree GBS groups

Key Result

Theorem 1

For fixed $p, q \in \mathbb{Z}$ with $|q| \geq |p|$ and $|p| \neq 1$, consider the group $\operatorname{BS}(p,q) = \langle x, t \mid x^p = tx^qt^{-1} \rangle$.

Figures (1)

  • Figure 1: The subgroup $\operatorname{Aut}^T(G)$ is the maximal subgroup of $\operatorname{Aut}(G)$ containing $\operatorname{Inn}(G)$ making the diagram commute, where $\theta$ is the canonical homomorphism with $\theta(G)=\operatorname{Inn}(G)$.

Theorems & Definitions (65)

  • Theorem 1: Corollary \ref{['cor:baumslagsolitar']}
  • Theorem 2: \ref{['thm:characterisingFg']}
  • Theorem 3: \ref{['cor:allFiniteOrder']}
  • Theorem 4: \ref{['thm:bounded']}
  • Theorem 5: \ref{['thm:treeFg']}
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • ...and 55 more