## Uniformization of symmetric Riemann surfaces by Schottky groups |

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2g circles anti-Mb'bius transformation Beltrami equation bounded by n closed analytic Jordan compatible conformal map cross caps Denoting by F diametrically opposed points DIASYMMETRIC SURFACES disjoint circles f holes f transition fundamental domain symmetric genus g group G homeomorphism identified pairs interior Koebe Theorem leaves fixed map f map of C-L map of C-L)/G maps the exterior Mobius transformations model of type multiply connected plane non-degenerate continua ORTHOSYMMETRIC pairs of circles pairs of Jordan pairs of quasicircles pairs of symmetrically plane domain bounded quasicircle is Identified quasiconf ormal QUASICONFORMAL MAPPINGS represented by reflection respect to reflection Schottky group situated Jordan curves situated with respect standard fundamental domain standard model surface of genus surface of type surface with f symmetric of type symmetric Riemann surface symmetric surface symmetric with respect symmetrically situated circles symmetrically situated Jordan symmetry t o transition curves type g unit circle Cn