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

TL;DR

This paper develops a boundary-action framework for Bass-Serre theory, translating graph-of-groups decompositions into dynamical actions on Bass-Serre trees and their boundaries. By connecting boundary freeness and strong boundary actions to $C^*$-simplicity, it produces new $C^*$-simple groups (notably certain tubular groups and $ ext{Out}(BS(p,2p))$) and pure infiniteness for the associated crossed products, yielding Kirchberg algebras satisfying the UCT under amenable actions. The authors recover known results for $ ext{GBS}_1$ groups and extend $C^*$-simplicity to a broad family of $ ext{GBS}_n$ groups, including the Leary-Minasyan example, and provide new, highly transitive examples. They apply the boundary-action approach to reduced graphs of groups, acylindrically hyperbolic vertex groups, and tubular groups, illustrating significant expansions in the landscape of non-amenable groups with simple or purely infinite $C^*$-algebras arising from boundary dynamics. Overall, the work offers concrete, boundary-driven criteria for $C^*$-simplicity and pure infiniteness, with broad implications for the structure and classification of the resulting crossed-product 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 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.

Figures (6)

• Figure 1: A non-singular but not reduced graph of groups
• Figure 2: Example of collapse and expansion moves.
• Figure 3: Tubular 2-Rose graph: $G_v\cong \mathbb{Z}\times \mathbb{Z}\cong \langle a,b\ |\ [a,b]=1 \rangle$, $G_e=\langle x \rangle\cong \mathbb{Z}$ where the monomorphism $\alpha_e: x\mapsto a^{m_1}b^{n_1}$ and $\alpha_{\bar{e}}: x\mapsto a^{m_2}b^{n_2}$ and $G_f=\langle y\rangle \cong \mathbb{Z}$ where $\alpha_f: y\mapsto a^{k_1}b^{l_1}$ and $\alpha_{\bar{f}}: y\mapsto a^{k_2}b^{l_2}$.
• Figure 4: The graph of groups $\mathbb{G}$ for $\mathrm{Out}(BS(p,q))$.
• Figure 5: The new graph of groups $\mathbb{G}_1$ for $\mathrm{Out}(BS(p,q))$.

Theorems & Definitions (137)

• Theorem A: Theorem \ref{['thm: reduced graph C star simple']}
• Theorem B: Theorem \ref{['thm: tubular']}
• Corollary 1: Corollary \ref{['cor: wise BB']}
• Theorem C: Theorem \ref{['thm: reduced graph c star simple 2']}
• Theorem D: Theorem \ref{['thm: out bs C simple']}
• Theorem E: Corollary \ref{['cor: gbs2 including LM']}
• Corollary 2
• Corollary 3
• Definition 2.1
• Definition 2.2