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

2025/02/04 by Ma, Xin, Wang, Daxun, Yang, Wenyuan · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2502.02039

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 Out(BS(p, q)) of Baumslag-Solitar groups. In addition, we study n-dimensional Generalized Baumslag-Solitar (GBS_n) groups. We first recover a result by Minasyan and Valiunas on the characterization of C^*-simplicity for GBS₁ groups and identify new C^*-simple 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.

Cited by

• Boundary actions of CAT(0) spaces: topological freeness and applications to C^*-algebras

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