Kazhdan's Property (T)
The Kazhdan ‘property (T)’ is a certain property of locally compact groups based on the existence of invariant vectors in their continuous unitary representations. The property has been used as a tool to demonstrate that a large class of its lattices is finitely generated. Chapter 1 gives an introduction to the subject, Chapter 2 concentrates on a parallel (and for many groups equivalent) property related to fixed points of the action of the group by affine isometries on a Hilbert space. Chapters 3 and 4 are focused on examples of non-compact groups with ‘property (T)’. Chapter 5 is an account of a spectral criterion for ‘property (T)’. Chapter 6 contains a small sample of applications (including a construction of expanders and an estimation of spectral gaps of operators, interesting from the point of view of ergodic theory). Chapter 7 is a short collection of open problems.