Synopsis: This course focuses on some of the most striking applications and developments of the theory of amenable groups and of the Alternative Theorem of Jacques Tits. Attendance of last year's TCC course with the same title is neither a requirement nor a hindrance: the overlap reduces to the starting point of this course, an overview of the notion of amenability and of the Alternative Theorem.

The course then continues with the Polynomial Growth Theorem of M. Gromov, its historical context and its proof, in which J. Tits' Alternative Theorem plays a major part. The Polynomial Growth Theorem is one of the first results stating that algebraic properties of an infinite group can be reconstructed from loose geometric features. Its proof contains a wealth of ideas that generated in their turn new areas of research.

Amenability also became meaningful in the investigation of actions of infinite groups on various types of spaces (Hilbert, Banach, non-positively curved). The last part of the course will focus on this development of the notion of amenability, in particular on connections with Kazhdan's property (T), Haagerup property (also called a-T-menability), and versions of these for Banach spaces and CAT(0) spaces. Proofs of the main results will be provided as time permits.

The plan of the lectures is as follows.

• Amenable groups, Jacques Tits' Alternative Theorem
  
  
• M. Gromov's Polynomial Growth Theorem: general context, statement and proof.
  
  
• Weaker and stronger versions of amenability. Fixed point properties and proper actions on Hilbert and Banach spaces, and on non-positively curved spaces.