the center of amalgamated product of free groups

Question

Let $G_1$, $G_2$, $H$ be free groups, $K=G_1*_H G_2$ is the amalgamated product of free groups, then is center of $K$ trivial? Thanks in advance.

Answer 1

Not necessarily. Take $\langle a, b \mid a^2 = b^2\rangle$. Then the centre is $\langle a^2\rangle$. But, if both $G_{1}$ and $G_{2}$ are non-abelian, then yes, because the centre of $K$ is $Z(G_{1})\cap Z(G_{2})\cap H$.