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On acylindrical tree actions and outer automorphism group of Baumslag-Solitar groups

Bratati Som, Daxun Wang

TL;DR

The paper investigates how acylindrical hyperbolicity behaves under quotients for groups acting on trees and identifies a largest acylindrical action for graphs of groups using Bass-Serre theory. It proves that quotients by an equivariant family of subgroups preserve $1$-acylindricity and provides concrete criteria guaranteeing non-elementary acylindrical quotients, thereby preserving acylindrical hyperbolicity. A general existence theorem for a largest acylindrical action is established in the setting of finite graphs of groups, with corollaries for finite vertex groups. As an application, the outer automorphism group of non-solvable Baumslag-Solitar groups, Out$(BS(p,q))$, is shown to be acylindrically hyperbolic, and the two natural tree actions arising from Clay's graph-of-groups structures are contrasted, revealing that the action on the edge-based tree is the largest acylindrical action.

Abstract

This paper explores acylindrical actions on trees, building on previous works related to the mapping class group and projection complexes. We demonstrate that the quotient action of a $1$-acylindrical action of a group on a tree by an equivariant family of subgroups remains $1$-acylindrical. We establish criteria for ensuring that this quotient action is non-elementary acylindrical, thus preserving acylindrical hyperbolicity of the group. Additionally, we show that the fundamental group of a graph of groups admits the largest acylindrical action on its Bass-Serre tree under certain conditions. As an application, we analyze the outer automorphism group of non-solvable Baumslag-Solitar groups, we prove its acylindrical hyperbolicity, highlighting the differences between various tree actions and identifying the largest acylindrical action.

On acylindrical tree actions and outer automorphism group of Baumslag-Solitar groups

TL;DR

The paper investigates how acylindrical hyperbolicity behaves under quotients for groups acting on trees and identifies a largest acylindrical action for graphs of groups using Bass-Serre theory. It proves that quotients by an equivariant family of subgroups preserve -acylindricity and provides concrete criteria guaranteeing non-elementary acylindrical quotients, thereby preserving acylindrical hyperbolicity. A general existence theorem for a largest acylindrical action is established in the setting of finite graphs of groups, with corollaries for finite vertex groups. As an application, the outer automorphism group of non-solvable Baumslag-Solitar groups, Out, is shown to be acylindrically hyperbolic, and the two natural tree actions arising from Clay's graph-of-groups structures are contrasted, revealing that the action on the edge-based tree is the largest acylindrical action.

Abstract

This paper explores acylindrical actions on trees, building on previous works related to the mapping class group and projection complexes. We demonstrate that the quotient action of a -acylindrical action of a group on a tree by an equivariant family of subgroups remains -acylindrical. We establish criteria for ensuring that this quotient action is non-elementary acylindrical, thus preserving acylindrical hyperbolicity of the group. Additionally, we show that the fundamental group of a graph of groups admits the largest acylindrical action on its Bass-Serre tree under certain conditions. As an application, we analyze the outer automorphism group of non-solvable Baumslag-Solitar groups, we prove its acylindrical hyperbolicity, highlighting the differences between various tree actions and identifying the largest acylindrical action.
Paper Structure (14 sections, 32 theorems, 33 equations, 8 figures)

This paper contains 14 sections, 32 theorems, 33 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Bass-Serre Theory
  3. Graph of groups
  4. Fundamental group of graph of groups
  5. Bass-Serre tree
  6. Action on the Bass-Serre tree
  7. Acylindrical tree actions
  8. Preserving acylindricity under quotient
  9. Largest acylindrical action of graph of groups
  10. Outer automorphism group of Gpq
  11. Deformation space of trees
  12. Two graph of groups structures of OutBSpq
  13. Quotient action of OutGpq
  14. Comparing tree actions of OutGpq

Key Result

Theorem A

Let $G$ be a group acting on a simplicial tree $T$, and let $N$ be a normal subgroup of $G$. If $N$ is a normal subgroup of $G$ generated by an equivariant family of subgroups $\{R_v\}$, then the quotient space $T/N$ is a tree. Moreover, suppose the action $G\curvearrowright T$ is a $(1,C)$-acylindr

Figures (8)

  • Figure 1: Two examples of graphs. The left graph consists of one edge with different extremities, whereas the right graph consists of one edge with the same extremities.
  • Figure 2: Example of collapse and expansion moves.
  • Figure 3: The graph of groups structure $(\Gamma_1,\mathcal{G}_1)$ of Out$(\hbox{BS}(p,q))$.
  • Figure 4: The graph of groups structure $(\Gamma_2,\mathcal{G}_2)$ of Out$(\hbox{BS}(p,q))$.
  • Figure 5: A local picture of the tree $T_{4,12}$. Every translate of $A$ has valence 4 and every translate of $B$ has infinite valence. The two red lines represent Axis($\psi\phi_i$) and Axis($\psi\phi_j$) respectively.
  • ...and 3 more figures

Theorems & Definitions (82)

  • Theorem A
  • Corollary B
  • Corollary C
  • Theorem D
  • Theorem E
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • ...and 72 more