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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.

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

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