Kristian Knudsen Olesen ANALYTICASPECTSOF THETHOMPSON GROUPS PhD Thesis Department of Mathematical Sciences University of Copenhagen PhD thesis in mathematics c Kristian Knudsen Olesen, 2016 Typeset in Times Roman and Computer Modern using LATEX. Printed by Polyteknisk Boghandel. ISBN 978-87-7078-948-6 PhD thesis submitted to the PhD School of Science, Faculty of Science, University of Copenhagen, Denmark in July 2016. Academic advisors: Magdalena Musat Uffe Haagerup University of Copenhagen, University of Southern Denmark, Denmark Denmark Assessment committee: Nicolas Monod Tullio Ceccherini- Ryszard Nest (chair) École Polytechnique Silberstein University of Copenhagen, Fédérale de Lausanne Universita del Sannio, Denmark Switzerland Italy Kristian Knudsen Olesen Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 DK-2100 København Ø Denmark
[email protected] I always suppose anything people please, [...] — Edmond Dantes, The Count of Monte Cristo Abstract In this thesis we study various analytic aspects of the Thompson groups, sev- eral of them related to amenability. In joint work with Uffe Haagerup, we prove that the Thompson groups T and V are not inner amenable, and give a criteria for non-amenability of the Thompson group F . More precisely, we prove that F is non-amenable if the reduced group C∗-algebra of T is simple. Whilst doing so, we investigate the C∗-algebras generated by the image of the Thompson groups in the Cuntz algebra O2 via a representation discovered by Nekrashevych. Based on this, we obtain new equivalent conditions to F being non-amenable. Furthermore, we prove that the reduced group C∗-algebra of a non-inner amenable group possessing the rapid decay property of Jolissaint is simple with a unique tracial state.