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...


1. Are there any interesting known restrictions on *amenable* Kähler groups?
2. 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.


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



Kahler groups with no non-abelian free groups?
For 1: beyond the case of nilpotent/solvable groups, I know of no restrictions on amenable kähler groups.


For 3: this is an interesting question but again I know of no restriction. This is probably due to our lack of ideas or to the lack of any suitable technology.


For 2: besides the 2-step nilpotent Kähler groups (not virtually Abelian) built in the 90s (Campana, Carlson-Toledo), I know of no new examples. In particular I think that it is still unknown whether there can be some 3-step nilpotent examples (not virtually 2-step nilpotent).


