Joint work with Joshua Frisch, Yair Hartman, and Pooya Vahidi Ferdowsi.
The Poisson boundary of a random walk on a group captures the uncertainty in the walk's asymptotic behavior. It has long been known that all commutative groups are Choquet-Deny groups: namely, they have trivial Poisson boundaries for every random walk. More generally, it has been shown that all virtually nilpotent groups are Choquet-Deny. We know 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.