On applications of the theory of random walks to the rigidity of lattices.

The theory of random walks on groups provides us a universal object associated to the group - the Poisson boundary. The Poisson boundary has some very special ergodic properties. In particular it is amenable and double ergodic. The Poisson boundary, as well as these properties will be discussed in Kaimanovich's talks.

It was highlighted in the work of Burger and Monod that the mere existence of such a well behaved space associated to any group has non-trivial corollaries.

In my talks I will illustrate this philosophy. I will discuss a general theory of the factors of the Poisson boundary, and explain how to exploit it in order to get rigidity results for lattices in groups. A particular case in which this approach is useful is of irreducible lattices in product of groups.

My talk will be based on an ongoing joint project with Alex Furman.