Andrei Malyutin (Steklov Institute of Mathematics, St. Petersburg)
  Random walks on groups acting on treelike spaces

Random walks on groups is a subject at the intersection of probability and group theory. We are looking for new examples and approaches in this area by examining group actions on topological spaces and applying methods of topological dynamics. The talk focuses on new observations related to random walks on groups acting on spaces with treelike structure. This class of spaces includes classical trees, R-trees, dendritic spaces, dendrons, dendrites, etc. For such spaces, we will discuss a series of results of the following kind.

Theorem. Let G be a countable group acting on a dendrite D. (A dendrite is a compact connected locally connected metric space containing no simple closed curve.) Assume that D is not an interval. Assume moreover that D contains no proper G-invariant connected compact subsets. Take a point x in D such that the set D\{x} is disconnected, and let f be the map from G to D sending g to g(x). Let m be a probability measure on G whose support generates G as a semigroup. Then f maps almost all paths of the (right) random walk with distribution m to convergent sequences.