TITLE: Cohomological dimension of knit products
QUESTION [10 upvotes]: Let $G$ be a group with complementary subgroups $A$ and $B$ (meaning $A\cap B=1$ and $AB=G$).
If $A$ and $B$ are both normal in $G$, then $G\cong A\times B$ is a direct product. If $A$ is normal, then $G\cong A\rtimes_{\varphi} B$ is a semi-direct product. More generally, $G\cong A\bowtie B$ is a knit product, or Zappa-Szép product, or general product of $A$ and $B$.
My question concerns the cohomological dimension of such knit products. To ask a specific question:

Does anyone know of any examples where $\operatorname{cd}(A\bowtie B)<\operatorname{cd}(A\times B)$? 

(Edit: Other than free products $A\ast B$.)
More generally I would be interested in any references concerning the cohomology of knit products. Is there any weaponry akin to the Lyndon-Hochschild-Serre spectral sequence?

REPLY [3 votes]: First of all, the free product of nontrivial groups is never a knit product of these groups. 
Secondly, a similar question for short exact sequences of groups was already asked in this MO post, with examples similar to Dieter's. 
The third remark is that, in view of the examples, the natural inequality to expect is subadditivity:
$$
cd(A\bowtie B) \le cd(A) + cd(B). 
$$ 
I do not know how to prove this inequality in general (apart from semidirect products). Something might follow from this paper where the related concept of "factorable group" is considered and the author did some homological computations (I did not look at the details though). 
Suppose, however, that instead of cohomological dimension $cd(G)$ of groups, one considers the asymptotic dimension $asdim(G)$ (I assume from now on, that both groups $A$ and $B$ are finitely generated). Definitions and basic results can be found, for instance here. It then follows from the "Hurewicz theorem" for asymptotic dimension, using the product map 
$$
A\times B \to A\bowtie B
$$
that
$$
asdim(A\bowtie B) \le asdim(A) + asdim(B).
$$