Program

• Amenablity of groups and actions: defintions and examples
• The Reiter and Folner conditions
• The Banach–Tarski paradox and Tarski’s theorem
• Amenability as a fixed point property
• Kesten’s characterization of amenability
• The Rokhlin lemma of Ornstein–Weiss
• Orbit equivalence relations
• Hyperfinite equivalence relations
• Dye’s theorem
• Non-amenable subgroups of PSL(2,R).

Exercises

• Exercise Sheet 1
• Exercise Sheet 2
• Exercise Sheet 3

Bibliography

• Wagon, The Banach–Tarski paradox, Cambridge University Press, 1985.
• Ceccherini-Silberstein, Grigorchuk, de la Harpe, Amenability and paradoxical decompositions for pseudogroups and discrete metric spaces, arXiv:1603.04212.
• Kerr, Li, Ergodic Theory: Independence and dichotomies, Springer, (Sections 4.5, 4.6 and Appendix A).
• Kechris, Miller, Topics in orbit equivalence, Springer.

Notes

• Dye’s theorem