TY - BOOK

T1 - Analytical aspects of the Thompson Groups

AU - Olesen, Kristian Knudsen

PY - 2016

Y1 - 2016

N2 - In this thesis we study various analytic aspects of the Thompson groups, severalof them related to amenability. In joint work with Uffe Haagerup, we provethat the Thompson groups T and V are not inner amenable, and give a criteria fornon-amenability of the Thompson group F. More precisely, we prove that F isnon-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 groupsin the Cuntz algebra ∂2 via a representation discovered by Nekrashevych. Basedon this, we obtain new equivalent conditions to F being non-amenable.Furthermore, we prove that the reduced group C'-algebra of a non-inneramenable group possessing the rapid decay property of Jolissaint is simple witha 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 constructone-parameter families of representations of the Thompson group F onthe Hilbert space L2([0; 1];m), where m denotes the Lebesgue measure, and weinvestigate when these are irreducible and mutually inequivalent. In addition,we exhibit a particular family of such representations, depending on parameterss ∈ R and p ∈ (0; 1), and prove that these are irreducible for all values of sand p, and non-unitarily equivalent for different values of p. We furthermoreshow that these representations are strongly continuous in both parameters, andthat they converge to the trivial representation, as p tends to zero or one.

AB - In this thesis we study various analytic aspects of the Thompson groups, severalof them related to amenability. In joint work with Uffe Haagerup, we provethat the Thompson groups T and V are not inner amenable, and give a criteria fornon-amenability of the Thompson group F. More precisely, we prove that F isnon-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 groupsin the Cuntz algebra ∂2 via a representation discovered by Nekrashevych. Basedon this, we obtain new equivalent conditions to F being non-amenable.Furthermore, we prove that the reduced group C'-algebra of a non-inneramenable group possessing the rapid decay property of Jolissaint is simple witha 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 constructone-parameter families of representations of the Thompson group F onthe Hilbert space L2([0; 1];m), where m denotes the Lebesgue measure, and weinvestigate when these are irreducible and mutually inequivalent. In addition,we exhibit a particular family of such representations, depending on parameterss ∈ R and p ∈ (0; 1), and prove that these are irreducible for all values of sand p, and non-unitarily equivalent for different values of p. We furthermoreshow that these representations are strongly continuous in both parameters, andthat they converge to the trivial representation, as p tends to zero or one.

UR - https://soeg.kb.dk/permalink/45KBDK_KGL/1pioq0f/alma99122190972905763

M3 - Ph.D. thesis

BT - Analytical aspects of the Thompson Groups

PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen

ER -