Bounded harmonic functions on groups are closely related to random walks on groups. It has long been known that all abelian groups, and more generally, virtually nilpotent groups are "Choquet-Deny" that is, cannot support non-trivial bounded harmonic functions. Equivalently, every random walk on such groups has a trivial Furstenberg-Poisson boundary.
I will present a recent result where we complete the classification of discrete countable Choquet-Deny groups, proving a conjuncture of Kaimanovich-Vershik. We show that any finitely generated group which is not virtually nilpotent, is not Choquet-Deny. Surprisingly, the key here is not the growth rate, but rather the algebraic infinite conjugacy class property (ICC).
This is joint work with Joshua Frisch, Omer Tamuz and Pooya Vahidi Ferdowsi.