Abstract
A method is presented for constructing closed surfaces out of Euclidean polygons with infinitely many segment identifications along the boundary. The metric on the quotient is identified. A sufficient condition is presented which guarantees that the Euclidean structure on the polygons induces a unique conformal structure on the quotient surface, making it into a closed Riemann surface. In this case, a modulus of continuity for uniformising coordinates is found which depends only on the geometry of the polygons and on the identifications. An application is presented in which a uniform modulus of continuity is obtained for a family of pseudo-Anosov homeomorphisms, making it possible to prove that they converge to a Teichmüller mapping on the Riemann sphere.
Citation
André de Carvalho. Toby Hall. "Paper folding, Riemann surfaces and convergence of pseudo-Anosov sequences." Geom. Topol. 16 (4) 1881 - 1966, 2012. https://doi.org/10.2140/gt.2012.16.1881
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