Let $U,U'\subseteq\mathbf{C}$ be two connected open sets such that $\pi_1(U)\simeq\pi_1(U')$.

Q: Does this imply that $U$ is homeomorphic to $U'$?

In the case where the $\pi_1$'s are trivial then the answer is yes, this is a consequence of Riemann's mapping theorem. May be one should try to prove it when the $\pi_1$'s are free groups on $n$ generators...