# Percolation on transitive graphs

May 5 to May 9, 2008

at the

American Institute of Mathematics, San Jose, California

organized by

Gabor Pete and Mark Sapir

## Original Announcement

This workshop will be devoted to percolation on transitive graphs, most importantly, on Cayley graphs of finitely generated infinite groups. Geometric properties of Cayley graphs often turn out to have counterparts in the probabilistic world, and vice versa, but the translations between the different viewpoints are not always trivial. The aim of this workshop is to bring together people working in geometric group theory, probability and dynamics to learn from each other about the relevant techniques in these fields and thus generate new momentum to solve some of the persistent open problems.

Some specific problems we would like address:

• Universality of critical percolation behavior: the conjectures regarding pc<1, pc<pu, pu<1, quasi-isometry invariance, critical exponents, the role of asymptotic cones. Could scaling limits be described analogously to SLE?
• What is the relation between lace expansion and the triangle condition of percolation theory, the rapid decay property, and hyperbolicity of groups?
• On what groups is renormalization possible?
• Relation of group theoretic properties (Kazhdan's property T, L2 and bounded cohomology, cost of groups, finite presentability) to percolation.
• Survival of geometric and random walk properties under percolation.
• The role of unimodularity in percolation.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

New examples of finitely presented groups with strong fixed point properties
by  Indira Chatterji and Martin Kassabov
Is the critical percolation probability local?
by  Itai Benjamini, Asaf Nachmias, and Yuval Peres
On $k$-free-like groups
by  A. Yu. Olshanskii and M. V. Sapir
The triangle and the open triangle
by  Gady Kozma