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 $d$.)