# Amenable groups with finite classifying space

A group $G$ is said to be elementary amenable if it can be obtained from finite and abelian groups by subgroups, quotients, extensions and increasing unions. It is well-known that all such groups are amenable, i.e. allow for a finitely additive and $G$-invariant probability measure on $G$. Grigorchuk's group is an example of a finitely generated group which is amenable but not elementary amenable. Grigorchuk has also found a finitely presented example of such a group in An example of a finitely presented amenable group not belonging to the class EG, 1998 Sb. Math. 189 75.

However, since Grigorchuk's example

$$\langle a,c,d,t \mid a^2 =c^2 =d^2 =(ad)^4 =(adacac)^4 =e, a^t = aca, c^t = dc, d^t = c \rangle$$

obviously contains torsion it cannot have a finite classifying space.

Question: Is there an example of an amenable group which is not elementary amenable and whose $BG$ is homotopy equivalent to a retract of a finite $CW$-complex?

Equivalently:

Question: Is there an example of an amenable group $G$, such that $G$ is not elementary amenable and the trivial $G$-module $\mathbb Z$ has a finite resolution by finitely generated projective $\mathbb ZG$-modules.

These kind of questions are sometimes called Day's problem for a certain class of groups.