I have the following question:
Let $G=A\underset{C}\star B$ be the free product of two groups $A$ and $B$ with amalgam $C$, such that $C\cap vCv^{-1}=1$ for all reduced words $v$ in G with length $\geq k$. Let $w\in G$ be a reduced word of length $\geq 2k+1$. I want to show that $\langle A,wAw^{-1} \rangle\cong A\star A$.
How can i show that $A\cap wAw^{-1}=1$? Thanks for help!