The Poisson boundary and the infinite conjugacy class property

The Poisson boundary and the infinite conjugacy class property

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Omer Tamuz, California Institute of Technology
Fine Hall 214

 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. I will present a recent result showing 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 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.