Pcmi Lecture Notes on Property (t ), Expander Graphs and Approximate Groups (preliminary Version)

The final aim of these lectures will be to prove spectral gaps for finite groups and to turn certain Cayley graphs into expander graphs. However in order to do so it is useful to have some understanding of the analogous spectral notions of amenability and Kazhdan property (T ) which are important for infinite groups. In fact one important aspect of asymptotic group theory (the part of group theory concerned with studying the geometric and group theoretic properties of large finite groups) is the ability to pass from the world of infinite groups to the that of finite groups and vice-versa and to manage to transfer results from one world to the other.