Choquet-Deny groups and the infinite conjugacy class property
Joshua Frisch, Yair Hartman, Omer Tamuz, Pooya Vahidi Ferdowsi

TL;DR
This paper characterizes when countable discrete groups are Choquet-Deny, showing finitely generated groups are so if and only if they are virtually nilpotent, and providing conditions involving quotients and conjugacy classes.
Contribution
It provides a complete characterization of Choquet-Deny groups, linking this property to virtual nilpotency and the infinite conjugacy class property.
Findings
Finitely generated Choquet-Deny groups are exactly the virtually nilpotent groups.
A countable group is Choquet-Deny iff none of its quotients has the infinite conjugacy class property.
Non-Choquet-Deny groups have symmetric, finite entropy, non-degenerate measures witnessing this.
Abstract
A countable discrete group is called Choquet-Deny if for every non-degenerate probability measure on it holds that all bounded -harmonic functions are constant. We show that a finitely generated group is Choquet-Deny if and only if it is virtually nilpotent. For general countable discrete groups, we show that is Choquet-Deny if and only if none of its quotients has the infinite conjugacy class property. Moreover, when is not Choquet-Deny, then this is witnessed by a symmetric, finite entropy, non-degenerate measure.
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