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.

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Choquet-Deny 群和无限共轭类属性

一个可数离散群$G$被称为Choquet-Deny，如果对于$G$上的每一个非退化概率测度$\mu$，它认为所有有界$\mu$-调和函数都是常数。我们证明有限生成群$G$是Choquet-Deny当且仅当它几乎是幂零的。对于一般可数离散群，我们证明 $G$ 是 Choquet-Deny 当且仅当它的商都没有无限共轭类属性。此外，当 $G$ 不是 Choquet-Deny 时，这可以通过对称、有限熵、非退化测度来证明。