# Dichotomy Amenable/Nonamenable in Combinatorial Group Theory

October 8 to October 12, 2007

at the

American Institute of Mathematics, Palo Alto, California

organized by

Tatiana Nagnibeda and Mark Sapir

## Original Announcement

This workshop will be devoted to various incarnations of the notion of amenability for a finitely generated group. The main goal of the workshop is to gain better understanding of the meaning of being amenable or nonamenable for a discrete, finitely generated group. Our attention will be concentrated on a certain number of concrete open problems about (non)amenability of groups with origins in very different areas of mathematics, and the workshop will bring together researchers from these diverse areas.

The main topics of the workshop are the following. We also indicate some concrete questions which we are going to discuss during the workshop.

1. Algebraic, geometric and probabilistic structure of amenable and nonamenable groups (is R. Thompson's group $F$ amenable? which groups generated by finite automata are amenable?);
2. amenability of Golod-Shafarevich groups (Vershik's question: Is there a discrete amenable Golod-Shafarevich group?);
3. amenability and unitarizability of uniformly bounded representations (Dixmier's Unitarizability Problem);
4. amenability and percolation (The conjecture of Benjamini and Schramm on non-unicity of percolation);
5. asymptotic invariants of Cayley graphs of amenable and non-amenable groups (what kind of growth of Foelner sets in an amenable group is possible? can an asymptotic cone of a non-virtually cyclic amenable group have a cut point?).

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

### Lecture notes

On Amenability of Group Algebras by Bartholdi (slides do not define terms which were presented on the board)

Dixmier’s Problem on Amenability by Pisier