Let $G$ be an HNN-extension $\langle a,t\mid t^{-1}a^2t=a^2\rangle $. Then I am going to show that the normal closure of $t$ intersects the center of $G$ trivially.
In the first step by using normal form, I already showed that $Z(G)=\langle a^2\rangle$. The next step is show that $\langle a^2\rangle $ intersects trivially $N$, where $N$ is the normal closure of $\langle t\rangle$.
Any suggestion for the second step?
Hint. Kill $t$ (that is, add the relation $t=1$ into the presentation). What is the order of $a$ in the image?
Hint to applying the hint. If $N\cap \langle a^2\rangle$ is non-trivial then the image of $a$ has finite order in $G/N$ (why?).