I have a question.

Let $\displaystyle G=\langle b,t | t^{-1}b^{\beta}tb^{\beta} \rangle$ be a HNN extension of base group $\displaystyle \langle b \rangle$ with stable letter $\displaystyle t$.

Is it true that $\displaystyle \langle b \rangle \cap \langle t \rangle=1$?

I think is it true.