Strongly Amenable Groups, Choquet-Deny Groups, and the Infinite Conjugacy Class Property

This thesis consists of two main parts. In the first part, we study a space of symbolic dynamical systems for countable discrete ICC groups and show that minimal proximal actions in that space are generic. This study leads to a characterization of countable discrete strongly amenable groups; a countable discrete group is strongly amenable if and only if it has no ICC quotients. In the second part, we show that a countable discrete group is Choquet-Deny if and only if it has no ICC quotients, where a group is called Choquet-Deny if the Poisson boundary of every non-degenerate measure on the group is trivial. Combining the aforementioned results, we get that a countable discrete group is Choquet-Deny if and only if it is strongly amenable. In the case of finitely generated groups, by an old result due to McLain (1956) and Duguid and McLain (1956) and our classifications, we see that strongly amenable groups and Choquet-Deny groups are the same as virtually nilpotent groups.