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

TL;DR

This paper investigates acylindrical actions on trees, establishing criteria for non-elementary acylindrical actions and demonstrating the acylindrical hyperbolicity of outer automorphism groups of Baumslag-Solitar groups.

Contribution

It introduces new criteria for quotient actions to remain acylindrical and applies these to analyze the outer automorphism groups of Baumslag-Solitar groups, proving their acylindrical hyperbolicity.

Findings

Quotient of a 1-acylindrical action remains 1-acylindrical.

Criteria for non-elementary acylindrical actions are established.

Outer automorphism groups of non-solvable Baumslag-Solitar groups are acylindrically hyperbolic.

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.