On the number of relations in free products of abelian groups

We consider the finitely generated groups constructed from cyclic groups by free and direct products and study the question of the smallest number of relations for a given system of generators. This question is related to the relation gap problem. We prove that if $m$ and $n$ are not coprime then the group $H_{m,n}=(\mathbb Z_m\times\mathbb Z)*(\mathbb Z_n\times\mathbb Z)$ cannot be defined using three relations in the standard system of generators. We obtain a similar result for the groups $G_{m,n}=(\mathbb Z_m\times\mathbb Z_m)*(\mathbb Z_n\times Z_n)$. On the other hand, we establish that for coprime $m$ and $n$ the image of $H_{m,n}$ in every nilpotent group is defined using three relations.