I Amenability
1 The Hausdorff Banach Tarski Paradox
1.1 Equidecomposability
1.2 Paradoxical Sets and Groups
1.3 The Sphere
1.4 Summary
1.5 More on F2 inside SO (3)
2 The Ping-Pong Lemma
2.1 First Formulation
2.2 Second Formulation
3 Amenability
3.1 Definition
3.2 Means
3.3 Banach Spaces
3.4 Asymptotically Invariant Nets
3.5 Følner Nets
3.6 Cayley Graphs
3.7 Back to Amenability
3.8 The Følner Condition
3.9 Abelian Groups
3.10 Summary
4 Application of Amenability
4.1 Banach Limits
4.2 Von Neumann Ergodic Theorem
5 Some Group Theory
5.1 Growth
5.2 Nilpotent Groups
6 Actions of Amenable Groups
6.1 Affine Actions
6.2 Actions on Compact Hausdorff Spaces
7 Elementary Amenably Groups
7.1 The Class of Elementary Amenable Groups
7.2 Subgroups
7.3 Quotient Groups
7.4 Extensions
7.5 Direct Limits
8 Topological Groups
8.1 Definition
8.2 Haar Measure
8.3 Amenability of Topological Groups
II Kazhdan’s Property (T)
1 A Bit of History
2 Property (T)
2.1 Definition and First Results
2.2 Induced Representation
2.3 Proof of Theorem II.2.1.13
2.4 Expanders
3 Property (FH)
3.1 Isometric Actions
3.2 Property (FH)
3.3 Back to Property (T)
3.4 Hyperbolic Space
4 SL(2,Qp) and its tree
4.1 The p-adic numbers Qp
4.2 The tree of SL(2,Qp)
5 Actions on Manifolds
5.1 The Jordan Theorem
5.2 Bierberbach Theorem
5.3 The Kazhdan-Margulis Theorem