Omer Tamuz, Caltech
"The Poisson boundary, strong amenability and the infinite conjugacy class property"
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. It easily follows from this that virtually nilpotent groups are also Choquet-Deny. I will present a very recent result showing that in the class of finitely generated groups, only virtually nilpotent groups are Choquet-Deny. More generally, a countable discrete group is Choquet-Deny if and only if it has no quotients with the infinite conjugacy class property (ICC). In another recent result we show that (countable discrete) groups with no ICC quotients are precisely those that are strongly amenable. Strong amenability is (a priori) a very different group property, defined by a group's topological actions, and a deeper connection between these two properties (strong amenability and Choquet-Deny) has yet to have been found.
This is joint work with Joshua Frisch, Yair Hartman and Pooya Vahidi Ferdowsi.