There are well-known results about nilpotent and solvable (=virtually nilpotent) Kähler groups coming from the work of (to name a few) Campana, Carlson-Toledo, Arapura-Nori, Delzant...

- Are there any interesting known restrictions on
*amenable*Kähler groups? - What about interesting known examples?

More generally,

**Definition.** A group $G$ is a **von-Neumann group** if it does not contain a non-abelian free group.

It is clear that any amenable group is a von-Neumann group.

- What is known about Kähler groups which are also von-Neumann group?