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Let us call a finitely generated group $G$ cohomologically rich if for each $k \geq 0$, we can find a subgroup $G'$ and a prime $p$ such that $H^k(G';\mathbb F_p) \neq 0$. Examples which come to mind are:

$\bullet$ Groups that have torsion: For $p \mid n$, we have $H^k(\mathbb Z/n; \mathbb F_p) \neq 0$ for all $k$.

$\bullet$ Groups that contain an infinite rank free abelian subgroup, as $H^k(\mathbb Z^k;\mathbb F_p) \neq 0$.

On the other hand, counterexamples are free abelian groups of finite rank and - more generally - fundamental groups of aspherical finite-dimensional CW-complexes.

Can someone give an example of a group which is finitely generated, torsionfree and does not contain an infinite rank free abelian subgroup, but still it is cohomologically rich?

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    $\begingroup$ Take the free product $H=\ast_{k=1}^\infty \mathbb{Z}^k$, and embed it in a $2$-generated group $G$ using the original method of Higman-Neumann-Neumann. Then $G$ is an HNN-extension of $H*F_2$ over two free subgroups. Clearly $G$ is cohomologically rich. The facts that it is torsion-free and has no subgroups isomorphic to $\mathbb{Z}^\infty$ follows from the standard theory of HNN-extensions. $\endgroup$
    – Ashot Minasyan
    Commented May 21, 2015 at 13:00

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