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

TL;DR

This paper introduces a method to construct closed Riemann surfaces from Euclidean polygons with infinite identifications, establishing conditions for conformal structure and analyzing convergence of pseudo-Anosov sequences to Teichmüller mappings.

Contribution

It provides a new construction technique for Riemann surfaces from polygons and demonstrates convergence properties of pseudo-Anosov homeomorphisms to Teichmüller maps.

Findings

Established a sufficient condition for conformal structure on quotient surfaces.

Derived a modulus of continuity for uniformizing coordinates.

Proved convergence of pseudo-Anosov sequences to Teichmüller mappings.

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 Teichm\"uller mapping on the Riemann sphere.