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 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 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 uniforraization 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 X.
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