*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](https://www.semanticscholar.org/paper/Non-positive-curvature-and-complexity-for-finitely-Bridson/d47a57a60cf5602ee193065546a0e420195d2a69) 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 $d$.)



Constraints on the homology of amenable groups

> 
> Is every finitely presented torsion-free amenable group of homotopical type F?
> 
> 
> 


It's false even for metabelian groups.


For $i=1,2,3$, let $G\_i$ be a copy of a solvable Baumslag-Solitar group, say $\langle t\_i,x\_i|t\_ix\_it\_i^{-1}=x\_i^2\rangle$. Start with $H=G\_1\times G\_2\times G\_3$. It has a homomorphism $\phi$ onto $\mathbf{Z}$ mapping each $t\_i$ to $1$ and each $x\_i$ to $0$. Let $G$ be its kernel. Then $G$ is finitely presented, but not of type FP$\_3$ / F$\_3$.


This follows from the "FP$\_m$-conjecture" for metabelian groups. Namely $H$ is not finitely presented by the Bieri-Strebel criterion [3], and $H$ is not FP$\_3$ / F$\_3$, as the "FP$\_m$-conjecture" is known for $m=3$ in a semidirect product of two abelian groups [2], and also known in general when the Prüfer rank is finite [1].


Note: the exponent $2$ could be replaced with any number $\ge 2$, possibly depending on $i$. Also, $\phi\_i(t\_i)$ could be defined as any positive integer $n\_i$, with the same conclusion (while if $\phi(t\_1),\phi(t\_2)>0$ and $\phi(t\_3)<0$ the resulting kernel has type F).




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[1] H. Åberg, Bieri–Strebel valuations (of finite rank), Proc. London Math. Soc. (3) 52 (1986) 269–304.


[2] R. Bieri, J. Harlander, On the FP3-conjecture for metabelian groups, J. London Math. Soc. (2) 64 (2001)
595–610.


[3] R. Bieri, R. Strebel, Valuations and finitely presented metabelian groups, Proc. London Math. Soc. (3) 41
(1980) 439–464.


