In my manuscript, we assume that $F_2=\langle a, b\rangle$ is free group. Also, we assume that $\varphi:F_2\times X\to X$ is generated by two homeomorphism $\varphi_a=f$ and $\varphi_b=g$.
Referee ask me why the group generated by two homeomorphism $f, g$ is free group?
What can conditions on $f, g$ to implies that the group generated by two homeomorphism $f, g$ is free group?