20.14 (2022)

(a) Do there exist a variety $\mathfrak{V}$ of groups and a group $G \in \mathfrak{V}$ such that the coproduct in $\mathfrak{V}$ of two copies of $G$ is embeddable in $G$, but the coproduct of three such copies is not? See (G. M. Bergman, Indag. Math., 18 (2007), 349–403).

Given an embedding $G \ast_{\mathfrak{V}} G \to G$, one might expect the induced map
$$G \ast_{\mathfrak{V}} (G \ast_{\mathfrak{V}} G) \to G \ast_{\mathfrak{V}} G \to G$$ to be an embedding. But this is not automatic, because in a general group variety $\mathfrak{V}$, a map $G \ast_{\mathfrak{V}} A \to G \ast_{\mathfrak{V}} B$ induced by an embedding $A \to B$ is not necessarily again an embedding.

(b) If there exist $\mathfrak{V}$ and $G$ as in (a), does there in fact exist an example with $\mathfrak{V}$ the variety of groups generated by $G$? For this and related questions, see (G. M. Bergman, Algebra Number Theory, 3 (2009), 847–879).