Flat surfaces in Sage

Introduction

For general mathematic references see the Flat surfaces wiki. A flat surfaces can be seen either

• as a union of polygons glued along pairs of parallel sides,
• as a flat metric with no holonomy on a surface,
• as a Riemann surface and a non zero Abelian (or quadratic) differential.

This page is aimed to be a roadmap for the implementations of various algorithm related to flat surfaces and more generally geometry of surfaces.

General architecture

For now the main structure is as follows

• sage.combinat.flat_surfaces
• sage.combinat.flat_surfaces.iet (for interval exchange transformations stuff)
• sage.combinat.flat_surfaces.origamis (for origamis/square tiled surfaces stuff)

Roadmap

Different representations/implementations for flat surfaces

• (convex) polygonal surface
  • rectangulated surface
    • suspension of iet (and li) (almost in Sage)
    • Thurston-Veech construction
  • triangulated surface
    • Delaunay surface (?)
• Algebraic curve with Abelian or quadratic differential
• Coverings (make it relative)... need to implement maps between translation surfaces
  • square tiled surfaces/origamis (covering of the torus) (almost in Sage)
  • hyperelliptic curves (specifying the double cover over the sphere)
• Unfoldings of rational billiards

Needed generic methods

• switch between representations (the one where everybody can be converted is triangulated flat surface)
• computing fundamental group and homology
• maps between flat surfaces
• action of SL(2,R) and isomorphisms

Hyperbolic geometry

• the three 2D models: hyperbolic plane HH, hyperbolic disc DD and the hyperboloïd

• polygonal domains
• tesselations (covering of HH by finite area convex polygonal domains)
• Fuchsian groups and fundamental domains