Property (T) for groups and von Neumann algebras

Specialeforsvar ved Tenna Nielsen

Titel: Property (T) for groups and von Neumann algebras

 Abstract: In this thesis we introduce property (T) for discrete groups, and use a proof of Yehuda Shalom's to prove that SL_n(Z ) has property (T), for n≥ 3. Hereafter, we study property (T) fornite von Neumann algebras, and prove that a group G has property (T) if and only if the group von Neumann algebra L(G) has property (T). Moreover, based on results due to Alain Connes' we prove that a type II₁ factor has countable outer automorphism group, and use this fact to prove that a type II₁ factor with property (T) has countable fundamental group. In the last part of the thesis we use Bachir Bekka's results to give a classification of finite factor representations of SL_n(Z), for n≥ 3. Additionally, we prove that C*(SL_n(Z)) does not have a faithful trace, for n≥ 3. 

Vejleder:   Mikael Rørdam
Censor:     Søren Møller, SDU