Choquet-Deny groups and the infinite conjugacy class property

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 $G$ 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.