$\DeclareMathOperator\cd{cd}$A question that I have already posted in the *Mathematics* section, but which seems to be too delicate for that section (see [here](https://math.stackexchange.com/questions/2320997/cohomological-dimension-of-direct-product) and [here](https://math.stackexchange.com/questions/2325286/cohomological-dimension-of-g-times-g)):


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](https://mathoverflow.net/questions/75472/kuenneth-formula-for-group-cohomology-with-nontrivial-action-on-the-coefficient). 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?



Cohomological dimension of $G \times G$
$\DeclareMathOperator\cd{cd}\newcommand{\FP}{\mathrm{FP}}$Another class of groups for which $\cd(G\times G)=2 \cd(G)$, containing duality groups, is the class of groups of type $\FP\_{\infty}$.


If $G$ is of type $\FP\_{\infty}$ that is the trivial module has a projective resolution that is finitely generated in each degree, then a theorem of Ken Brown [https://www.math.cornell.edu/~kbrown/scan/1975.0050.pdf](https://www.math.cornell.edu/%7Ekbrown/scan/1975.0050.pdf) implies that the functor $H^n(G,-)$ commutes with direct limits for all $n$. Thus if $G$ has cohomological dimension $k$ then there is a finitely generated $G$-module $A$ with $H^k(G,A)\neq 0$ (there is some module for which this cohomology doesn't vanish and any module is a direct limit of its finitely generated submodules). So your Kunneth argument should give $\cd(G\times G)=2 \cd(G)$ if $G$ is of type $\FP\_{\infty}$.


This includes the example of duality groups by Corollary 1 of Brown's paper although his argument is essentially Mark Grant's.


