Say I have a reduced word of the form $\alpha * \beta \in \pi_1(A) *_{\pi_1(A \cap B) }\pi_1(B)$ , where $\alpha \in \pi_1(A)$ and $\beta \in \pi_1(B)$.
Now what is the multiplication of $\alpha*\beta$ supposed to represent ? Is it the concatenation of $\alpha$ and $\beta$ ?
Note: I am not asking for the technical details of the definition of an almagamated product, I think I understand it. I am asking for the interpretation of $*$ in this particular context (if there exists one).