Take an orientable surface $S_g^s$ of genus $g$ with no boundaries and $s$ points removed and fix a complete hyperbolic metric of finite area (assuming that the Euler characteristic allows an hyperbolic structure on $S$). We know from hyperbolic geometry that there exists a unique geodesic in every non trivial free-homotopy class of curves. I know for sure that there is at least one point on $S_g^s$ through which there is no simple geodesic, but I have no idea about how to prove it. Could you help me with that?
Thank you!