Ergodic theorems for Markvian groups

Ordinary ergodic theorems (the Birkgoff and fon Neumann) theorems can be easily generalied to amenable groups. We are going to discuss an opposite example of groups: the groups that look like free groups. More precise we are going to discuss the Markovian groups. For actions of such groups in probabilty space one can obtain several generalization of ergodic theorems dealing with convergence for averages over spheres, balls and and convergence of averages over balls in Chesaro's sence.
These generalizations will be discssed in the talk