$\DeclareMathOperator\cd{cd}$Are there any known examples of non-free groups with a property that $\cd(G)+1 = \cd(G \times G)$, or, less restrictive, $G, H$ with $\cd \neq 1, \infty$ such that $\cd(H)+1 = \cd(G \times H)$?



Groups with unusual cohomological dimension of direct product
Let $G=(\mathbb{Q},+)$. Then ${\rm cd}(G)=2$ and ${\rm cd}(G\times G)=3$.


