Séance Séminaire

Séminaire Gaston Darboux

Which groups have bounded harmonic functions?

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.