Paper folding, Riemann surfaces, and convergence of pseudo-Anosov sequences

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 uniformizing 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.

75 pages, 18 figures

Cited by in corpus (3)

• Wild translation surfaces and infinite genus
• Unimodal Measurable Pseudo-Anosov Maps
• Limits of sequences of pseudo-Anosov maps and of hyperbolic 3-manifolds