Poisson Boundary, Liouville Property and Asymptotic Geometry of Linear Groups

The Poisson-Furstenberg boundary  is a measure space that describes asymptotics of infinite trajectories of random walks. The boundary is non-trivial if and only if the defining  measure admits non-constant bounded harmonic functions.

The origin of the notion of Poisson boundary goes back to works of Blackwell, Feller and Doob, and in the case of random walks on groups to that of Dynkin-Malyutov and Furstenberg. Furstenberg (1963) provides a description of the boundary for semi-simple Lie groups (for measures absolutely continuous with respect to the Haar measure). An analogous description holds in discrete setting (Furstenberg; Kaimanovich).

The last few decades brought a signifiant progress in understanding of  Poisson boundaries of groups. However, in the very classical case of linear groups even a basic question of boundary triviality or non-triviality remains open. We will speak about  partial progress and  a new conjectural description of this property for linear groups, various phenomena  for the action on the boundary and the connection to other geometric properties of solvable linear groups. We  plan to  discuss many open problems about geometry of linear groups.