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Flat surfaces in Sage

Introduction

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

• as a union of polygons glued along pairs of parallel sides,
• as a flat metric with no holonomy on a compact surface,
• as a Riemann surface together with 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/combinatoric/dynamic of surfaces (Mapping class group, train track, pseudo-Anosov dynamic, ...). For the moment we share the sage-combinat repository with mercurial for the development.

General architecture

For now the main structure is as follows

• sage.combinat.flat_surfaces (which contains various generic objects)
• sage.combinat.flat_surfaces.iet (for interval exchange transformations stuff)
• sage.combinat.flat_surfaces.origamis (for origamis/square tiled surfaces stuff)

Roadmap

Port of other programs

• Joshua Bowman program on iso-Delaunay tessellations (written in Java)
• Finish Anton Zorich port of Interval Exchange Transformations and Linear Involutions (written in Mathematica)
• Anton Zorich program for computing approximation of various Lyapunov exponents (written in C and Mathematica)
• Alex Eskin program for analyzing saddle connections direction in a surface (written in C++)

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 a double cover of the sphere)
• Unfoldings of rational billiards

Needed generic methods

• switch between representations (the one to which everybody can be converted is triangulated flat surface)
• computing fundamental group and homology
• maps between flat surfaces
• action of SL(2,R) and isomorphisms
• Siegel Veech constants
• Lyapunov exponents

Hyperbolic geometry

This part is roughly implemented in trac #9439

• the three 2D models: hyperbolic plane, hyperbolic disc and the hyperboloïd
• points, geodesics and polygonal domains
• tesselations (covering of HH by finite area convex polygonal domains)
• Fuchsian groups, their fundamental domains and their associated tesselations