Random walks on dense subgroups

Imagine you have a group, with a discrete subgroup. Wouldn’t that be nice to relate random walks, and Poisson boundaries of the group and of the subgroup, in a meaningful way? This was done by Furstenberg for lattices in semisimple Lie groups as an essential tool in an important rigidity result. We are concerned with dense subgroups. We develop a technique for doing it that allows us to exhibit some new interesting phenomena in Poisson boundary theory. I’ll explain the setting in which we work, and will focus mainly on our construction (leaving the applications as “further reading”). Joint work with Michael Björklund and Hanna Oppelmayer