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 present a combinatorial description of these origamis.Origami curves are certain special Teichmüller curves in some moduli space of curves. They are obtained from an unramified covering of a once punctured torus, see Section 1.1 for a precise definition.First examples of Teichmüller curves were already given by Veech in [V]. In recent years they have attracted a lot of attention, partly because of their relation to rational billiards, see e.g. [McM] and references therein; a survey of examples defining primitive Teichmüller curves can be found in [HuSc]. Another topic of interest is the action of the absolute Galois group of the rationals on them as discussed in [Lo] and [M1].In some respects Teichmüller curves arising via origamis are more accessible than general ones. Precisely for them, the Veech group (as defined in Section 1.1) is a subgroup of SL 2 (Z), cf. [GJ]. It can be determined for each origami explicitly, cf. [S]. Nevertheless, it is still difficult to approach the question how their Teichmüller curves are located in the moduli space. There are only a few origami curves for which explicit equations have been found so far, cf. [H] and [M1].In this note we present an extraordinary origami curve in genus 3. It is the smallest nontrivial example of a normal origami having as Veech group the full group SL 2 (Z), see Prop. 2. The Jacobian of the associated family of curves has a two-dimensional fixed part, see Proposition 7. M. Möller has observed that this implies that our origami curve is also a Shimura curve; recently he proved that it is the only algebraic curve in a moduli space of curves of genus at least 2 which is at the same time a Teichmüller curve and a Shimura curve, cf.[M2]. An explicit equation for the associated family of curves is given in Proposition 5. This family of curves has been studied by several authors, and some of the results of Section 1 have been known previously, cf.[Gi], [Gu2], [KK]; nevertheless the use of the "origami" structure allows for new proofs and makes the exposition considerably more elementary.A very remarkable property of this origami curve, and the main new result of this paper, is the fact that it intersects infinitely many other origami curves, see Theorem 1 in Section 3. To our knowledge this is the first example of origami curves that intersect in moduli space.The decomposition of the Jacobian gives a second map onto an elliptic curve; if this map is ramified only over torsion points it can be made into an origami. In Section 2 we develop explicit formulas for these maps to determine when this particular type of ramification occurs.We give a combinatorial description of the infinitely many origamis that intersect our ori...

We study the Veech group of an origami, i.e. of a translation surface, tessellated by parallelograms. We show that it is isomorphic to the image of a certain subgroup of Aut + (F 2 ) in SL 2 (Z) ∼ = Out + (F 2 ). Based on this we present an algorithm that determines the Veech group.

We study the boundary of Teichmüller disks in T g , a partial compactification of Teichmüller space, and their image in Schottky space. We give a broad introduction to Teichmüller disks and explain the relation between Teichmüller curves and Veech groups. Furthermore, we describe Braungardt's construction of T g and compare it with the Abikoff augmented Teichmüller space. Following Masur, we give a description of Strebel rays that makes it easy to understand their end points on the boundary of T g . This prepares the description of boundary points that a Teichmüller disk has, with a particular emphasis to the case that it leads to a Teichmüller curve. Further on we turn to Schottky space and describe two different approaches to obtain a partial compactification. We give an overview how the boundaries of Schottky space, Teichmüller space and moduli space match together and how the actions of the diverse groups on them are linked. Finally we consider the image of Teichmüller disks in Schottky space and show that one can choose the projection from Teichmüller space to Schottky space in such a manner that the image of the Teichmüller disk is a quotient by an infinite group.

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