Amenable groups with finite classifying space

Question

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.

Answer 1

There are very few examples of amenable but not elementary amenable finitely presented groups. There exists a torsion-free example, an ascending HNN extension of the basilica group (Bartholdi, Laurent; Virág, Bálint, Amenability via random walks. Duke Math. J. 130 (2005), no. 1, 39–56 - there it is proved that the basilica group is amenable but not elementary amenable; and R. I. Grigorchuk and A. Z˙ uk, On a torsion-free weakly branch group defined by a three state automaton, Int. J. Algebra Comput. 12(1–2) (2002) 223–246, R. I. Grigorchuk and A. ˙Zuk, Spectral properties of a torsion-free weakly branch group defined by a three state automaton, in Computational and Statistical Group Theory (Las Vegas, NV/Hoboken, NJ, 2001), Contemporary Mathematics, Vol. 298 (American Mathematical Society, Providence, RI, 2002), pp. 57–82 - there it is proved that this group has a finitely presented ascending HNN extension). Essentially this is the only known example (there are several other similar examples, see references in Bartholdi, Laurent, Eick, Bettina, Hartung, René A nilpotent quotient algorithm for certain infinitely presented groups and its applications. Internat. J. Algebra Comput. 18 (2008), no. 8, 1321–1344. ). I doubt very much that it has a finite classifying space or even has a finite projective resolution with finitely generated modules, because the basilica group itself is infinitely presented.