On Oikawa?s theorem from an algebraic and geometric point of view

Abstract

In 1956 K. Oikawa proved that a bordered compact Riemann surface X of genus g with k boundary components can be embedded into a closed Riemann surface \( \widetilde{X} \) of the same genus in such a way that its complement consists in a disjoint union of k discs and every automorphism of X extends to an automorphism of \( \widetilde{X} \) . Much later, in 1982, N. Greenleaf and C. L. May mention that the analytical arguments of Oikawa can be extended to the case of nonorientable compact surfaces. Here we give a new algebraic proof, based on the uniformization theorem, of a similar result for Riemann and Klein surfaces, together with a geometric interpretation that relates the geometry of fundamental regions for groups uniformizing the surfaces X and \( \widetilde{X} \) .