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. We then provide some applications of this criteria. In the last part of the thesis, inspired by recent work of Garncarek, we con- struct one-parameter families of representations of the Thompson group F on the Hilbert space L2([0, 1], m), where m denotes the Lebesgue measure, and we investigate when these are irreducible and mutually inequivalent. In addition, we exhibit a particular family of such representations, depending on parame- ters s R and p (0, 1), and prove that these are irreducible for all values of s ∈ ∈ and p, and non-unitarily equivalent for different values of p. We furthermore show that these representations are strongly continuous in both parameters, and that they converge to the trivial representation, as p tends to zero or one. Resumé I denne afhandling studerer vi diverse analytiske aspekter af Thompson-grup- perne, flere af hvilke er relateret til amenabilitet. I samarbejde med Uffe Haagerup viser vi, at Thompson-grupperne T og V ikke er indre amenable og giver et kri- terium, for at Thompson-gruppen F ikke er amenabel. Helt konkret viser vi, at F ikke er amenabel, hvis den reducerede gruppe-C∗-algebra hørende til T er simpel. I processen undersøger vi de C∗-algebraer som bliver genereret af billederne af Thompson-grupperne via en repræsentation i Cuntz-algebraerne O2, der blev op- daget af Nekrashevych. Baseret på dette giver vi nye ækvivalente betingelser for, at F ikke er amenabel. Derudover viser vi, at hvis en gruppe ikke er indre amenabel, men har ra- pid decay-egenskaben introduceret af Jolissaint, da er dens reducerede gruppe- C∗-algebra simpel og har en unik sportilstand. Efterfølgende præsenterer vi nogle anvendelser af dette resultat. Den sidste del af afhandlingen er inspireret af resultater af Garncarek. I denne konstruerer vi ét-parameter-familier af repræsentationer af Thompson-gruppen F på Hilbertrummet L2([0, 1], m), hvor m betegner Lebesguemålet. Vi undersøger, hvornår disse repræsentationer er irreducible, og hvornår de ikke er parvist ækvi- valente. Derudover producerer vi en konkret familie af sådanne repræsentationer, afhængig af to parametre s R og p (0, 1), samt viser at disse er irreducible ∈ ∈ for alle værdier af s og p, og ikke unitært ækvivalente for forskellige værdier af p. Ydermere viser vi, at repræsentationerne er stærkt kontinuerte i begge parametre og konvergerer mod den trivielle repræsentation, når p går med nul eller en. Contents Contents7 1 Introduction9 1.1 Notation and terminology........................ 12 2 The Thompson groups 15 2.1 The definition.............................. 15 2.2 Rearrangement of dyadic partitions................... 18 2.3 The Minkowski question mark function ................ 21 2.4 Piecewise projective linear maps.................... 29 3 Inner amenability 41 3.1 Amenable actions............................ 41 3.2 Introduction to inner amenability and property Γ . 49 3.3 The Thompson groups and inner amenability ............. 52 4 Operator algebras and the Thompson groups 55 4.1 A representation in the Cuntz algebra ................. 55 4.2 Simplicity and unique trace....................... 59 4.3 The rapid decay property........................ 66 4.4 A criterion for C∗-simplicity...................... 68 5 Families of irreducible representations 75 5.1 One map, lots of representations.................... 75 5.2 Ergodicity and equivalence relations.................. 86 5.3 Bernoulli measures on the unit interval................. 93 5.4 A construction..............................103 5.5 A re-construction............................115 5.6 Further thoughts and projects......................117 Bibliography 119 Index 125 7 Chapter 1 Introduction In 1965 Richard Thompson introduced three groups now commonly referred to as the Thompson groups. These groups, denoted by F , T and V , have been the cen- ter of much study since their introduction, and a particularly famous open problem regarding these groups asks whether or not F is amenable. This question has been studied intensely and several unsuccessful attempts have been made to prove or dis- prove amenability of F . The three groups satisfy F T V , and it is well-known ⊆ ⊆ that the Thompson groups T and V are non-amenable, as they contain a copy of F2, the free group on two generators. However, F does not contain a copy of F2, as proved by Brin and Squire [8] in 1985. Thus F has been for a long time a candidate for a finitely generated counterexample to the von Neumann conjecture, stating that the only obstruction to amenability is containment of F2. The von Neumann con- jecture was disproved in 1980 by Olshanskii [56], but his counterexample was not finitely generated. A finitely generated counterexample was found later by Olshan- skii and Sapir [58], and, recently, several highly accessible counterexamples have been produced by Monod [53] and by Lodha and Moore [50]. In this thesis we ap- proach the question of amenability of F by proving that the Thompson group F is non-amenable if the reduced group C∗-algebra of T is simple, thus giving an operator algebraic criterion for non-amenability of F . This result has been obtained in joint work with Uffe Haagerup, who was our main supervisor until his passing away in July, 2015. After announcing this result at various conferences, several partial con- verses have been obtained, first by Bleak and Juschenko [6] and later by Breuillard, Kalantar, Kennedy and Ozawa [7]. Recently, Le Boudec and Matte Bon [49] proved that the (full) converse statement holds, namely that F is non-amenable if and only if the reduced group C∗-algebra of T is simple. The study of groups whose reduced group C∗-algebra is simple, also known as C∗-simple groups, respectively, of groups whose reduced group C∗-algebra has a unique tracial state, also known as groups with the unique trace property, has been an important topic in operator algebra theory since 1975, when Powers [60] proved that F2 is C∗-simple and has the unique trace 9 10 CHAPTER 1. INTRODUCTION property. Over the years, many groups have been proven to be C∗-simple with the unique trace property, and a famous long time open question of de la Harpe asked whether or not C∗-simplicity and the unique trace property of a group are equivalent. Recently, a tremendous progress has been made regarding these two properties, and the question of de la Harpe has been completely settled. It was first proved by Poz- nansky [61] in 2009 that the two properties are equivalent for linear groups, however, it turns out that they are not equivalent in general. More precisely, in 2014 Kalan- tar and Kennedy [43] gave a characterization of C∗-simplicity in terms of certain boundary actions of the given group, and later the same year Breuillard, Kalantar, Kennedy and Ozawa [7] proved that C∗-simplicity implies the unique trace property. The year after, Le Boudec [48] gave an example of a C∗-simple group without the unique trace property, thus settling the question of de la Harpe. It is already known that the Thompson group T has the unique trace property, as shown by Dudko and Medynets [25] in 2012, but is remains an open question whether T is C∗-simple. In order to prove that C∗-simplicity of the group T implies non-amenability of F , we use a representation of the Thompson groups in the unitary group of the Cuntz algebra O2, introduced by Nekrashevych [55] in 2004. In addition to this result, we also prove, in collaboration with Haagerup, that the C∗-algebras generated by the representations of F , T and V are distinct, and that the one generated by V is all of O2. We then give equivalent conditions to F being non-amenable in terms of whether certain ideals in the reduced group C∗-algebras of F and T are proper. Another result we prove in collaboration with Haagerup is that the Thompson groups T and V are not inner amenable, thus settling a question that Chifan raised at a conference in Alba-Iulia in 2013. The notion of inner amenability was introduced by Effros [26] in 1975 in order to give a group theoretic characterization of property Γ of Murray and von Neumann for group von Neumann algebras of discrete ICC groups.