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

Abstract

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 $\mathrm{Out}(\mathrm{BS}(p,q))$ of Baumslag–Solitar groups. In addition, we study $n$-dimensional generalized Baumslag–Solitar (${\mathrm{GBS}}_n$) groups. We first recover a result by Minasyan and Valiunas on the characterization of $C^{\ast }$-simplicity for ${\mathrm{GBS}}_1$ groups and identify new $C^{\ast }$-simple ${\mathrm{GBS}}_n$ 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.