Let $G$ be a group. I am interested in the following property:
For any groups $A,B$ and monomorphism $G \hookrightarrow A \times B$, either $G \hookrightarrow A$ or $G \hookrightarrow B$.
For instance, simple groups satisfy this property. Other examples come from the following argument:
Suppose that $G$ does not satisfy our property, ie., there exists a monomorphism $G \hookrightarrow A \times B$ such that $G$ embeds neither in $A$ nor in $B$. Therefore, the kernel $G \cap A$ of the morphism $$G \hookrightarrow A \times B \twoheadrightarrow A$$ is not trivial. Similarly, $G \cap B$ must be non trivial. Thus, if $a \in G \cap A$, then its centralizer contains $\langle a \rangle \times (G \cap B)$. In particular, if $G$ is torsion-free, the centralizer of one of its element contains $\mathbb{Z}^2$.
Consequently, torsion-free hyperbolic groups (such as free groups) satisfy the property, as well as many right-angled Artin groups. Do you know other examples?
Nota Bene: I am pretty sure I already met this property elsewhere, so a reference should exist. If someone knows such a reference, he's welcome.