NOTES FROM PAST CLASSES

Advanced Class in Algebra 2013-14

Tuesdays 3-4:30 pm Room C5

Amenable and Sofic Groups

To start with, we will discuss the origins of amenability in measure theory and the Banach Tarski paradox. Our quick foray into analysis will involve Tarski's theorem about the equivalence of the existence of an invariant mean and the non-existence of a paradoxical decomposition. We will define amenability using invariant means and also, discuss the combinatorial approach using Folner sets. Two other definitions which may be of interest will be the one due to Kesten and the cohomological one.


From here, we will progress to examples and non-examples, closure theorems and consequences to group theory. For instance,

  1. finitely generated groups of subexponential growth are amenable.

  2. if a countable discrete group contains a (non-abelian) free subgroup on two generators, then it is not amenable. The converse to this statement is the von Neumann conjecture and was disproved by Olshanskii in 1980. We may look at some contemporary counterexamples to the conjecture.

The relatives referred to in the title are the sofic groups. Broadly speaking, sofic groups are groups which may be `approximated by finite groups' and in this respect, soficity is a sweeping generalisation of amenability. Gromov asks whether all finitely generated groups are sofic. Later in the year, we may turn to sofic groups, `for want of other idleness'!


Alejandra Garrido's Four Lectures on Amenability

Henry Bradford's Lecture on Ergodic Methods

Aditi Kar's Lecture on Non-amenable Torsion groups

Aditi Kar's Notes on Sofic Groups

References

  1. Alan Peterson's book Amenability

  2. Appendix G from Bekka and Vallette's book Kazhdan's property (T)

  3. Entry from Terry Tao's blog

  1. Chapter from Lubotzky's book Discrete groups, expanding graphs and invariant means

  2. Pestov's notes on Soficity