# 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 $\mu$ 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.

## Authors

Joshua Frisch

Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA

Yair Hartman

Department of Mathematics, Ben-Gurion University of the Negev, Be'er Sheva, Israel

Omer Tamuz

Division of the Humanities and Social Sciences and Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA

Pooya Vahidi Ferdowsi

Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA