Recently I've been reading about cohomological finiteness conditions for groups, my main source being Brown's book "Cohomology of Groups".

One of the first things one learns is that a group with finite cohomological dimension necessarily is torsionfree. It is easy to come up with an example of a torsionfree group with infinite cohomological dimension: non-finitely generated free abelian group immediately springs to mind. A little bit of googling revealed an example of an infinite-dimensional torsionfree $FP_{\infty}$ group.$^1$ (A group is said to be of type $FP_{\infty}$ if there exists a projective $ZG$-resolution {$P_i$} of $Z$ such that each $P_i$ is finitely generated.) This example is Thompson's group $F$. However, it is well-known that $F$ contains a non-finitely generated free abelian group.

So, the question is: what is an example of a torsionfree group with infinite cohomological dimension and no inifinitely generated free abelian subgroup (that is, if there exists one)?

$^1$ In case anyone is interested: K. S. Brown, R. Geoghegan, Invent. Math. 77, 367--381.