An extraordinary origami curve

  title={An extraordinary origami curve},
  author={Frank Herrlich and Gabriela Schmithuesen},
  journal={Mathematische Nachrichten},
We study the origami W defined by the quaternion group of order 8 and its Teichmüller curve C (W) in the moduli space M3. We prove that W has Veech group SL2(ℤ), determine the equation of the family over C (W) and find several further properties. As main result we obtain infinitely many origami curves in M3 that intersect C (W). We also present a combinatorial description of these origamis. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 
Teichmüller curves defined by characteristic origamis
We study translation surfaces with Veech group SL2(Z). They all arise as origamis; any characteristic subgroup of F2 provides an example. For any given origami, we construct one having Veech group
A comb of origami curves in the moduli space M3 with three dimensional closure
The first part of this paper is a survey on Teichmüller curves and Veech groups, with emphasis on the special case of origamis where much stronger tools for the investigation are available than in
Homology of origamis with symmetries
Given an origami (square-tiled surface) M with automorphism group G, we compute the decomposition of the first homology group of M into isotypic G-submodules. Through the action of the affine group
Veech groups and extended origamis
In this paper, we deal with flat surfaces of finite analytic type with two distinct Jenkins-Strebel directions. We show that such a flat surface is characterized by decomposition into parallelograms
Origamis associated to minimally intersecting filling pairs
Let Sg denote the closed orientable surface of genus g. In joint work with Huang, the first author constructed exponentially-many (in g) mapping class group orbits of pairs of simple closed curves
An algorithm for classifying origamis into components of Teichm\"uller curves
Non trivial examples of Veech groups have been studied systematically with the notion of combinatorics coming from coverings. For abelian origamis, coverings of once punctured torus, their Veech
A characterization of Veech groups in terms of origamis.
Schmithusen proved in 2004 that the Veech group of an origami is closely related to a subgroup of the automorphism group of the free group $F_2$. This result is significant in the sense that the
Totally non congruence Veech groups
Veech groups are discrete subgroups of SL(2, R) which play an important role in the theory of translation surfaces. For a special class of translation surfaces called origamis or square-tiled
Shimura- and Teichmueller curves
We classify curves in the moduli space of curves that are both Shimura- and Teichmueller curves: Except for the moduli space of genus one curves there is only a single such curve. We start with a
The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis
We compute explicitly the action of the group of affine diffeomorphisms on the relative homology of two remarkable origamis discovered respectively by Forni (in genus 3) and Forni-Matheus (in genus


An Algorithm for Finding the Veech Group of an Origami
The Veech group of an origami, i.e., of a translation surface, tessellated by parallelograms, is studied and it is shown that it is isomorphic to the image of a certain subgroup of Aut)+F 2 in SL2(Z) ≅ Out+(F 2).
Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards
SummaryThere exists a Teichmüller discΔn containing the Riemann surface ofy2+xn=1, in the genus [n−1/2] Teichmüller space, such that the stabilizer ofΔn in the mapping class group has a fundamental
A family of arithmetic surfaces of genus 3
The aim of this paper is the study of the genus 3 curves C n : Y 4 = X 4 - (4n - 2) X 2 + 1, from the Arakelov viewpoint. The Jacobian of the curves C n splits as a product of elliptic curves, and
Prime arithmetic Teichmuller discs in H(2)
It is well-known that Teichmuller discs that pass through "integer points'' of the moduli space of abelian differentials are very special: they are closed complex geodesics. However, the structure of
Teichmueller curves, Galois actions and GT-relations
Teichmueller curves are geodesic discs in Teichmueller space that project to algebraic curves $C$ in the moduli space $M_g$. Some Teichmuller curves can be considered as components of Hurwitz spaces.
Billiards and Teichmüller curves on Hilbert modular surfaces
This paper exhibits an infinite collection of algebraic curves isometrically embedded in the moduli space of Riemann surfaces of genus two. These Teichmuller curves lie on Hilbert modular surfaces
Invariants of translation surfaces
We definite invariants of translation surfaces which refine Veech groups. These aid in exact determination of Veech groups. We give examples where two surfaces of isomorphic Veech group cannot even
  • P. Lochak
  • Mathematics
    Journal of the Institute of Mathematics of Jussieu
  • 2005
We study several types of curves and higher-dimensional objects inside the moduli spaces of curves, insisting on their arithmetic properties in the perspective of Grothendieck–Teichmüller theory. On
Teichmüller curves, Galois actions and $ \widehat {GT} $‐relations
Teichmüller curves are geodesic discs in Teichmüller space that project to algebraic curves C in the moduli space Mg. Some Teichmüller curves can be considered as components of Hurwitz spaces. We
Affine mappings of translation surfaces: geometry and arithmetic
1. Introduction. Translation surfaces naturally arise in the study of billiards in rational polygons (see [ZKa]). To any such polygon P , there corresponds a unique translation surface, S = S(P),