Uniformization of Symmetric Riemann

1. Introduction. A Riemann surface S is called symmetric if there exists an anti-conformal map 0 of S onto itself such that <p2 = identity. We say that 0 is a symmetry on S. The classical "retrospection theorem" asserts the existence of representations of closed Riemann surfaces of genus g by "Schottky groups," groups generated by Möbius transformations Ax,-,Ag such that A; maps the exterior of i\ into the interior of rf, where Ti, T[, T'g are disjoint Jordan curves bounding a 2g-times connected domain, a standard fundamental domain for the group. We will show that a closed symmetric Riemann surface of genus g can be represented by a Schottky group which has a standard fundamental domain which exhibits the symmetry. This result is contained in Theorems I, II and III of § §4-6. The proof does not use the classical theorem. As a corollary, in §7, we obtain a new proof of the Koebe theorem: every n-times connected planar domain can be conformally mapped onto a plane domain exterior to n disjoint circles. Techniques from the theory of quasiconformal mappings are used to obtain these results. I would like to thank Professor Lipman Bers for his invaluable advice and assistance.