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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Paper folding, Riemann surfaces and convergence of pseudo-Anosov sequences

André de Carvalho and Toby Hall

Geometry & Topology 16 (2012) 1881–1966

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.

geometric structures on surfaces, Riemann surfaces, pseudo-Anosov sequences
Mathematical Subject Classification 2010
Primary: 30C35, 30F10, 37E30
Secondary: 30C62, 30F45, 37F30
Received: 18 July 2011
Revised: 5 April 2012
Accepted: 30 May 2012
Published: 27 August 2012
Proposed: Danny Calegari
Seconded: Leonid Polterovich, Yasha Eliashberg
André de Carvalho
Departamento de Matemática Aplicada, IME - USP
Rua do Matão 1010
Cidade Universitária
05508-090 São Paulo
Toby Hall
Department of Mathematical Sciences
University of Liverpool
Liverpool L69 7ZL