Colloquium[ Edit ]
Moderator: Ron Rosenthal
Given a countable group G and a probability measure m on G, a function from G to the reals is said to be m-harmonic if it satisfies the mean-value property with respect to averages taken using m. It has long been known that all commutative groups are Choquet-Deny groups: namely, they admit no non-constant bounded harmonic functions. More generally, it has been shown that all virtually nilpotent groups are Choquet-Deny. We show that in the class of finitely generated groups, only virtually nilpotent groups are Choquet-Deny. This proves a conjecture of Kaimanovich and Vershik (1983), who suggested that groups of exponential growth are not Choquet-Deny. Our proof does not use the superpolynomial growth property of non-virtually nilpotent groups, but rather that they have quotients with the infinite conjugacy class property (ICC). Indeed, we show that a countable discrete group is Choquet-Deny if and only if it has no ICC quotients.
Joint work with Joshua Frisch, Yair Hartman and Pooya Vahidi Ferdowsi