Amenable groups and Jacques Tits'
Alternative theorem

**Taught Course Centre (May-June 2014)**

**Time:**
Thursday 10-12.
The TCC timetable can be found here.

**Place:** Videoconference Room VC1, Mathematical Institute.
**Synopsis:**
I plan to cover the following topics, listed in the order of their appearance in lectures.

- Overview of the course. The Banach-Tarski Paradox.

- Amenable graphs: Cheeger constant, Gromov's condition, quasi-isometric
invariance.

- Amenable groups: equivalent definitions, stability under group operations,
the von Neumann-Day conjecture.

- Jacques Tits' Alternative Theorem for linear groups.

- If time permits: M. Gromov's Polynomial Growth Theorem with a sketch of proof.

The course is based on chapters of the book "Lectures on Geometric Group
Theory", written jointly with Misha Kapovich.
The corresponding chapters are available here.
The text is in a preliminary version, still undergoing changes, therefore it is
better not to print it in its current form. I have left all the preliminary chapters of the book in this text for reference,
not all are actually needed for this course.
Comments and corrections are most welcome.
**Lecture notes: **

Lecture 1: slides.

Lecture 1: whiteboard notes.

Lecture 2: slides.

Lecture 2: whiteboard notes.

Lecture 3: slides.

Lecture 3: whiteboard notes.

Lecture 3:diagram.

Lecture 4: slides.

Lecture 4: whiteboard notes.

Lecture 5: slides.

Lecture 5: whiteboard notes.

Lecture 6: slides.

Lecture 6: whiteboard notes.

Lecture 7: slides.

Lecture 7: whiteboard notes.

Lecture 8: slides.

Lecture 8: whiteboard notes.

Recommended reference for algebraic groups (unfortunately out of print): the book of Onishchik and Vinberg.