Boundary actions of Bass-Serre Trees and the applications to $C^*$-algebras

Abstract

In this paper, we study Bass-Serre theory from the perspectives of $C^*$-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 $C^*$-simple groups including certain tubular groups, fundamental groups of certain graphs of groups with one vertex group acylindrically hyperbolic and outer automorphism groups $\operatorname{Out}(BS(p, q))$ 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 $C^*$-simplicity for $\text{GBS}_1$ groups and identify new $C^*$-simple $\text{GBS}_n$ groups including the Leary-Minasyan group. These $C^*$-simple groups also provide new examples of $C^*$-selfless groups and highly transitive groups. Moreover, we demonstrate that natural boundary actions of these $C^*$-simple fundamental groups of graphs of groups give rise to the new purely infinite crossed product $C^*$-algebras.