Actions of solvable Baumslag-Solitar groups on hyperbolic metric spaces (Q6164264)

A solvable Baumslag-Solitar group has presentation \(\mathrm{BS}(1,n)=\langle a,t \mid a^{t}=a^{n} \rangle\) (\(\mathrm{BS}(1,1) \simeq \mathbb{Z} \times \mathbb{Z}\) and, for \(n \geq 2\), \(\mathrm{BS}(1,n) \simeq \mathbb{Z}[\frac{1}{n}] \rtimes \mathbb{Z}\) is a nonabelian solvable group). In this paper, the authors give a complete list of the cobounded actions of solvable Baumslag-Solitar groups on hyperbolic metric spaces up to a natural equivalence relation. The set of equivalence classes carries a natural partial order first introduced in [Algebr. Geom. Topol. 19, No. 4, 1747--1835 (2019; Zbl 1481.20165)] by the first author et al., and in this paper the resulting poset is completely described. There are finitely many equivalence classes of actions, and each equivalence class contains the action on a point, a tree or the hyperbolic plane.