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

We study non-ascending HNN extensions acting on their Bass– Serre tree 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. Résumé. — Nous étudions les extensions HNN non-ascendantes agissant sur leur arbre de Bass–Serre and caractérisons la C∗-simplicité et la propriété d’unicité de la trace à l’aide du noyau et des quasi-noyaux des extensions HNN en question. Nous présentons aussi un exemple concret d’extension HNN fournissant un nouvel exemple de groupe C∗-simple mais ne possédant pas la propriété d’unicité de la trace. De plus, nous obtenons certains resultats plus généraux, la plupart se fondant sur des travaux antérieurs d’auteurs divers, concernant la C∗-simplicité de groupes admettant des actions extrêmes sur des bords, et, en particulier, de groupes agissant sur des arbres

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

We study non-ascending HNN extensions acting on their Bass– Serre tree 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