Harmonic functions on groups and the Liouville property

Specialeforsvar: Norberto Clemente Delgado

Titel: Harmonic functions on groups and the Liouville property

Abstract: The overall interest of this thesis is the study of bounded harmonic functions on groups. We will start by discussing the classical complex analysis setup for the study of real-valued harmonic functions, their characterisation through the mean value property and how they relate with holomorphic functions and the Liouville theorem. This setup will motivate the definition of harmonic functions on groups and the discussion of the conditions for all bounded harmonic functions on a countable and discrete group G being necessarily constant. For this, we will first construct the Poisson boundary Πµ of a group G equipped with a Borel probability measure µ and discuss some of its applications. This boundary will be an auxiliary measure space that will allow us to represent any harmonic function on the group G via a bounded function on the boundary using that the corresponding Poisson transform Pµ is an isometric isomorphism. The construction uses random walks on the group. We then investigate conditions for the Poisson boundary being trivial, that is, for all bounded µ-harmonic functions on the group being constant (as in the classical case). Notions like amenability, entropy and growth of groups will play an important role in this
task. Finally, we will discuss the recent characterisation of the triviality of the Poisson boundary due to Frisch, Hartman, Tamuz and Ferdowski in their paper Choquet-Deny groups and the infinite conjugacy class property. Namely, the triviality of the Poisson
boundary will be characterised through the existence of ICC quotients in the group.

Vejleder: Magdalena Musat
Censor:   Andreas Midjord