Dichotomy Amenable/Nonamenable in Combinatorial Group Theory

This workshop, sponsored by AIM and the NSF, 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?).

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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