Analytic aspects of the Thompson groups

PhD defense: Kristian Knudsen Olesen

In this thesis we study various analytic aspects of the Thompson groups, several 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 O_2 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 construct one-parameter families of representations of the Thompson group F on the Hilbert space L^2([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 parameters s in \mathbb{R} and p in (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.

Supervisor:
Associate Prof. Magdalena Musat, MATH, University of Copenhagen

Assessment committee:
Prof. Ryszard Nest (chairman), MATH, University of Copenhagen
Prof. Nicolas Monod, Ecole Politechnique Federale de Lausanne (EPFL)
Prof. Tullio Ceccherini-Silberstein, Universita del Sannio