The Poisson boundary of a group has two interpretations. Firstly it measures the asymptotic uncertainty of a random walk on a group. Secondly it classifies the possible range of bounded harmonic functions on the group. In this minicourse we will introduce some of the theory of the Poisson boundary and its relationship with group properties such as polynomial growth, the infinite conjugacy class property, and amenability. In particular we will focus on the following question. For which measured groups are all bounded harmonic functions trivial? This will lead us into a, perhaps surprising, incredibly interconnected web of ideas including convex analysis, dynamical systems, information theory, and probability theory. No prior knowledge of random walks on groups or any of the aforementioned fields will be assumed.
These videos were part of the Geometric group theory without boundaries II virtual summer school.
