Choquet-Deny groups and the infinite conjugacy class property

Abstract

A countable discrete group G is called Choquet-Deny if for every non-degenerate probability measure μ on G, it holds that all bounded μ-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.

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