Analytic Topics in Group Theory

This is the webpage for my Part III course on Analytic Topics in Group Theory, which was lectured in Lent Term 2011.

Chapter 1
Introduction. Basic properties of growth in groups. Virtually nilpotent groups have polynomial growth.

Chapter 2
Jordan's theorem on finite subgroups of GL_n(C).

Chapter 3
The Banach-Tarski Paradox. Amenability.

Examples 1
Example Sheet 1

Examples 2
Example Sheet 2

The course did continue from here, containing a complete proof of Gromov's Theorem following Kleiner, then a proof of Varopoulos's result that the simple random walk is recurrent if and only if the group is a finite extension of 1,Z or Z^2. I started turning the notes into a book, which will contain somewhat more. However, I became distracted from that project (though I will return to it). It might be possible to extract drafts of the relevant sections from me.

Other resources:

Emmanuel Breuillard's notes on Jordan's original proof of Jordan's theorem .

Math Overflow article on free subgroups of SO(3).

Terry Tao's blog notes on Banach-Tarski etc.

Chapter 1 Introduction. Basic properties of growth in groups. Virtually nilpotent groups have polynomial growth.
Chapter 2 Jordan's theorem on finite subgroups of GL_n(C).
Chapter 3 The Banach-Tarski Paradox. Amenability.
Examples 1 Example Sheet 1
Examples 2 Example Sheet 2