Properties of Kazhdan- and Haagerup-type in the context of modern analytic group theory
We will provide a general overview about the goals and methods of analytic group theory and survey a few milestone results, which explain how analytic properties can be used to divide the universe of groups into different areas. We will then focus on two properties of groups, which are in some sense opposite extremes: Kazhdan’s Property (T) and the Haagerup property, also known as a-T-menability. We will discuss various different characterizations, variants and generalizations of these properties. We will also relate Property (T) to compact generation and actions on trees and the Haagerup Property to amenability.