This question arises from section $2$, exercise $13.8$ of Bogopolski's Introduction to Group Theory. I managed to show that $\text{GL}_2(\mathbb{Z})\cong D_4 *_{D_2} D_6$. Now, I want to take a random word in this HNN extension and find its normal form.
My question is:
I have $\text{GL}_2(\mathbb{Z})$ written as an amalgamated product, and I want to write it as an HNN extension. What to do?
I speculate that it is going to be something like $$\langle{a,b,t \mid t^{-1}a^2t=b^3}\rangle$$ like in example 14.2 (same section, same book), where the base group is $\text{SL}_2(\mathbb{Z})$ and the associated subgroups are $\langle{a^2}\rangle$ and $\langle{b^3}\rangle$. However, I am having trouble justifying this.