Poisson boundaries of groups: - Entropy, growth and amenability

Specialeforsvar ved: Andreas Midjord

The main interest of this thesis is the study of the Poisson boundary of a group G and its applications. The concept, originally introduced by Furstenberg in 1963, is based on the idea of extending the boundary representation of harmonic functions from the classical Dirichlet problem to a general group-theoretic setting. The Poisson boundary itself is an abstractly defined probability space, on which any harmonic function on G can be represented (using the Poisson transform, appropriately defined) by a boundary function on it. The Poisson boundary is associated to a random walk on the group, and the construction is done both in a topological and a measurable setting, where in the latter case two explicit realisations are provided.

We then investigate conditions ensuring triviality of the Poisson boundary and show that this can, in many cases, be characterised by means of vanishing of the Shannon entropy, a criteria due to Kaimanovich and Vershik. Further, we study applications of the Poisson boundary to analytic properties of the group, such as amenability, and discuss a related conjecture by Furstenberg. Finally, we tie in the notion of amenable actions, as defined by Zimmer. While an amenable group always acts amenably on any G-space, it is an important feature that any group acts amenably on its Poisson boundary. This is a result
of Zimmer, which we discuss.

As an additional treat, we mention very recent surprising applications of the Furstenberg boundary (which we also introduce) to the settling of a number of long-standing open problems regarding group C*-algebras.

Vejleder: Magdalena Musat

Censor: Alexander Sokol