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.

For examples using the algorithm developed here see dynamics/examples.

This project take part in the wider dynamics 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)
• sage.geometry.hyperbolic_geometry (hyperbolic spaces)
• sage.groups.surface_gps (abstract surface groups)

Where do we put

• representation of surface group into PSL(2,R)
• fundamental domains of such groups / Poincare polygons / Dirichlet fundamental domains

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 relative homology and homology (as well as functors between them)
• maps between flat surfaces (and functors to fundamental group and homologies)
• action of SL(2,R) and isomorphisms (and functors)
• Siegel Veech constants
• Lyapunov exponents

Surface Group

They are needed from two point of vue: the group of the surface itself and its stabilizer under SL(2,R) or PSL(2,R) action. There must be some software for dealing with surface group. We need to look at

• kbmag: Knuth-Bendix in Monoids and Automatic Groups implemented by Derek Holt

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
• tessellations (covering of HH by finite area convex polygonal domains)
• Fuchsian groups, their fundamental domains and their associated tessellations

The Experimental Geometry Lab (university of Maryland) published a lot of Mathematica package/worksheets to deal with Kleinian adn Fuchsian groups, hyperbolic tessellations, etc...