As the title suggests, my question is does $\mathbb{Z}\* \mathbb{Z} /N = \mathbb{Z} \* \mathbb{Z}$ imply that $N$ is trivial? Here, $\mathbb{Z} \* \mathbb{Z}$ is the free product of $\mathbb{Z}$ and $\mathbb{Z}$ and $N$ is some normal subgroup of $\mathbb{Z}\* \mathbb{Z}$. My intuition tells me that the answer is yes, despite this not holding for other groups; a quick example is $\mathbb{S}^1 = \{ e^{i\theta} \in \mathbb{C} \}$, since $\mathbb{S}^1/\{\pm 1\} = \mathbb{S}^1 $.

An equivalent problem would be, given a surjective epimorphism $\varphi: \mathbb{Z} \* \mathbb{Z} \to \mathbb{Z}\*\mathbb{Z}$, does $\varphi$ have to be injective as well?