Choquet-Deny groups and the infinite conjugacy class property

A countable discrete group \(G\) is called Choquet-Deny if for every non-degenerate probability measure \(\mu\) on \(G\) it holds that all bounded \(\mu\)-harmonic functions are constant. We show that a finitely generated group \(G\) is Choquet-Deny if and only if it is virtually nilpotent. For general countable discrete groups, we show that \(G\) is Choquet-Deny if and only if none of its quotients has the infinite conjugacy class property. Moreover, when \(G\) is not Choquet-Deny, then this is witnessed by a symmetric, finite entropy, non-degenerate measure.