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 conjecture 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.
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