Amenability, Liouville property and ergodicity of boundary actions.
Abstract. We will start with cogrowth criterion of amenability and explain how it was used to treat the alternative amenable/nonamenable in the case of discrete groups and how the first nontrivial estimations of Tarski number were obtained. Then the self-similar groups will come to the arena and we will explain why they and their Schreier graphs are useful in study of amenability and random walks. Then we will switch to topics around Poisson Boundary and Liouvulle property for groups and graphs and study the ergodic properties of boundary actions of a free group and of its subgroups. The Nielsen and Schreier-Redeimeister methods as well as cogrowth will be used for description of partitioning into conservative and dissipative parts.