*Edit (March 24): My first question has been answered nicely, but I am still looking for an answer to the second one.*

Due to the Kan–Thurston theorem, the homology of an arbitrary group can be anything you want.

Using the Rips complex, we can see that hyperbolic groups are $F_{\infty}$ and, if torsion-free, even of type $F$, i.e., there exists a model of $BG$ that is a finite CW-complex.

The other side of Bridson's universe of finitely presented groups is where amenable groups live. I wondered about constraints that amenablility imposes on the cohomology of a group. More precisely, let me ask the following two separate questions.

Let $G$ denote a finitely presented torsion-free amenable group.

- Is $G$ always of type $F$?

(True for nilpotent groups, see Brown's book. In general, this is probably false or open: Wikipedia taught me that it is an unresolved conjecture to prove that Thompson's group $F$ is not amenable.)

- Can it happen that $H_{j}(G;Z)$ is non-trivial in a single degree $d$?

(For $d = 1$, we can obviously choose $G = \mathbf{Z}$, but I do not know any examples for bigger $d$.)