$\DeclareMathOperator\cd{cd}$A question that I have already posted in the Mathematics section, but which seems to be too delicate for that section (see here and here):
Let $\cd(G)$ denote the cohomological dimension of a group $G$, i.e. the minimal length of a projective resolution of Z over $\mathbb{Z} G$. Under which conditions on a group $G$ of finite cohomological dimension is it true that $$\cd(G \times G) > \cd(G) \; ?$$ On the one hand, there are examples of groups with $G \times G \cong G$, though I am not aware of examples of finite cohomological dimension. On the other hand, an affirmative answer follows from the Kuenneth formula for group cohomology as discussed in Chris Gerig's detailed answer to this question. Nevertheless, this line of argument is only applicable if we find a local coefficient system $A$ for which $H_*(G;A)$ is finitely generated and for which there exists $k\in \mathbb{N}$ with $2k>\cd(G)$ and $H^k(G;A)\neq 0$. Under which conditions does such $A$ exist?