Conformal Transformations of Riemann Surfaces

k=1 defined by p(ak) = gk, p (bk) = gk-'- If K is the kernel, then G is isomorphic to F/K. F is the fundamental group of a compact Riemann surface . of genus n. - is covered by the half-plane R = { (x, y) I y > 0}, and the group of covering transformations is a discrete group of linear fractional transformations, isomorphic to F. If we identify points of .) which are congruent under K, we obtain a Riemann surface C- = /K. (E is compact, since it is a finite covering space of - == $/F (the index [F: K] = order (G) is finite). A conformal transformation c of e can be lifted to a coniformal transformation