Boundary actions of Bass-Serre Trees and the applications to $C^{\ast }$-algebras

In this paper, we study Bass-Serre theory from the perspectives of $C^{\ast }$-algebras and topological dynamics. In particular, we investigate the actions of fundamental groups of graphs of groups on their Bass-Serre trees and the associated boundaries, through which we identify new families of $C^{\ast }$-simple groups including certain tubular groups, fundamental groups of certain graphs of groups with one vertex group acylindrically hyperbolic and outer automorphism groups $\mathrm{Out}(BS(p,q))$ of Baumslag-Solitar groups. In addition, we study $n$-dimensional Generalized Baumslag-Solitar (${\text{GBS}}_n$) groups. We first recover a result by Minasyan and Valiunas on the characterization of $C^{\ast }$-simplicity for ${\text{GBS}}_1$ groups and identify new $C^{\ast }$-simple ${\text{GBS}}_n$ groups including the Leary-Minasyan group. These $C^{\ast }$-simple groups also provide new examples of $C^{\ast }$-selfless groups and highly transitive groups. Moreover, we demonstrate that natural boundary actions of these $C^{\ast }$-simple fundamental groups of graphs of groups give rise to the new purely infinite crossed product $C^{\ast }$-algebras.