Introduction to Property (T)

Abstract: We discuss different perspectives on Kazhdan’s property (T): As a property of unitary representations, as a fixpoint property, as a strong negation of amenability, as a spectral gap property etc. We then discuss various consequences concerning finite generation,  actions on trees and homomorphisms into abelian groups. Concerning examples, we sketch an argument which ensures that SL_n(Z) has Property (T) if and only if n is at least 3 and which can be generalized to arbitrary higher rank lattices.