Geometry & Topology 16 (2012) 1881–1966

DOI: 10.2140/gt.2012.16.1881

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.

Keywords

geometric structures on surfaces, Riemann surfaces, pseudo-Anosov sequences

Mathematical Subject Classification

Primary: 30C35, 30F10, 37E30

Secondary: 30C62, 30F45, 37F30

References

Publication

Received: 18 July 2011
Revised: 5 April 2012
Accepted: 30 May 2012
Published: 27 August 2012
Proposed: Danny Calegari
Seconded: Leonid Polterovich, Yasha Eliashberg

Authors