Abstract:
If a holomorphic map f: C→D of compact Riemann surfaces has no ramification points, then the induced map of their fundamental group is injective.
So for the surjectivity of the fundamental group of the induced map, it is necessary to ramify.. But being ramified alone does not imply surjectivity. Those maps where the induced map has the surjectivity of the fundamental group are called genuinely ramified. In this talk, we will show many equivalent conditions for genuine ramification. It is shown that the genuine ramification is equivalent to: