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 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.
Additional Information© 2019 Department of Mathematics, Princeton University. J. Frisch was supported by NSF Grant DMS-1464475. Y. Hartman was partially supported by the Israel Science Foundation (grant No. 1175/18). He is grateful for the support of Northwestern University, where he was a postdoctoral fellow when most of this research was conducted. O. Tamuz was supported by a grant from the Simons Foundation (#419427).
Submitted - 1802.00751.pdf