Synopsis: Kazhdan's Property (T) and Haagerup's Property (also called a-T-menability) are two important properties of infinite groups, relevant in many settings:

in combinatorics (finite quotients of infinite groups with property (T) are expanders);

in algebra (they imply structural properties of infinite groups);

in smooth dynamics (they imply rigidity results);

in ergodic theory;

in Operator Algebras (they display strong connections with the Baum-Connes conjecture)


In this course I shall overview some key facts about both properties. The course is based on chapters of the book "Geometric Group Theory", written jointly with Misha Kapovich. An older version of the book can be found here.

The lecture notes and exercise sheets are posted here:

Lecture 1.