Property tau and von Neumann algebras
Buettgens, Matthew Robert
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This dissertation investigates property τ for groups and its connection with finite von Neumann algebras, particularly group von Neumann algebras. The first chapter gives the motivation for this work and introduces properties T and τ for groups. The second chapter describes Hilbert modules and von Neumann correspondences. In the third chapter we summarize previous results defining rigidity and property T for von Neumann factors and for group von Neumann algebras. The fourth chapter begins our study of property τ. We begin by defining a much more general concept, uniform quasirigidity. We then define property τ for group von Neumann algebras and two related properties relevant to all finite von Neumann algebras. We show that a finitely-generated discrete group Γ has property τ if and only if its group von Neumann algebra has property τ according to our definition. We give a number of equivalent characterizations of our definitions. In the fifth chapter, we describe some fundamental constructions in dynamical systems and show that the properties we described in the previous chapter have important applications to these. The sixth chapter returns to group property τ. We describe a cohomological characterization of that property due to Lubotzky and Zuk. We then contrast it with a new characterization of property τ in terms of group actions on nonatomic probability spaces. We conclude with important questions for future research.