C*-simplicity of HNN extensions and groups acting on trees

TL;DR

This paper investigates the C*-simplicity and unique trace property of HNN extensions acting on trees, providing characterizations, examples, and general results related to groups with boundary actions.

Contribution

It offers new characterizations of C*-simplicity for HNN extensions and presents a novel example of a group with the unique trace property but not C*-simple.

Findings

Characterization of C*-simplicity via kernels and quasi-kernels.

Construction of a new HNN extension example with unique trace property.

General results on C*-simplicity for groups acting on trees.

Abstract

We study non-ascending HNN extensions acting on their Bass-Serre trees, and characterize C*-simplicity and the unique trace property by means of the kernel and quasi-kernels of the HNN extension in question. We also present a concrete example of an HNN extension that is a new example of a group that is not C*-simple but does have the unique trace property. Additionally, we include certain more general results, mostly based on previous work of various authors, concerning C*-simplicity of groups admitting extreme boundary actions, and in particular, groups acting on trees.