Amenable groups of finite cohomological dimension I seek an example of an amenable group of finite cohomological dimension that is not virtually polycyclic and has finite abelianization.
Remarks.


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*Elementary amenable groups of finite cohomological dimension are virtually solvable.

*There are lots of virtually abelian groups of finite cohomological dimension that have finite abelianization. The simplest one is the fundamental group of a certain flat 3-manifold, the Hantsche-Wendt manifold.

*Virtually polycyclic groups are finitely presented, so perhaps an example can be found among virtually solvable groups that are not finitely presentable, which do exist.

*Elementary amenable groups of cohomological dimension two are classified here, see Theorem 3, and they have infinite abelianization (except for the trivial group). 

*Another idea would be to search for an amenable group that is not elementary amenable.
There are very few known examples, and I do not know if any of the examples have finite cohomological dimension and finite abelianization.  See here for a list of torsion-free examples (of course, finite cohomological dimension implies torsion-free.
 A: Igor, what about doing the following. 
Let $A$ be a virtually polycyclic group of finite cohomological dimension whose abelianization is finite, but has has ${\mathbb Z}_n$, $n$ even, among its cyclic factors. Say, the fundamental group of a Hantsche-Wendt manifold would do the job. Next, take $B$, the semidirect product of $\mathbb Q$ and of ${\mathbb Z}$, where ${\mathbb Z}$ acts (through ${\mathbb Z}_2$) by $q\to -q$. Now, make $A$ act on $B$ where the action on $\mathbb Q$ is trivial and the action on ${\mathbb Z}$ is by $n\to -n$ (this is indeed an action as $\mathbb Z$ is abelian). Lastly, let $G$ the semidirect product of $B$ and $A$ using the above action. By construction $G$ has finite abelianization. Also $G$ has finite cohomological dimension because its semidirect factors have this property ($\mathbb Q$ is locally cyclic). Finally, $G$ is not virtually polycyclic as it contains $\mathbb Q$, and subgroups of virtually polycyclic groups are finitely generated. (This paragraph is edited by Igor Belegradek).
You can also imitate the construction of H-W groups, where ${\mathbb Z}_2^n$ would act 
on ${\mathbb Q}^n$ rather than ${\mathbb Z}^n$, but I did not think about the details. 
