Related papers: An extraordinary origami curve
We construct new coordinates for the Teichm\"uller space Teich of a punctured torus into $\bold{R} \times\bold{R}^+$. The coordinates depend on the representation of Teich as a space of marked Kleinian groups $G_\mu$ that depend…
This paper is devoted to the classification of the infinite families of Teichmuller curves generated by Prym eigenforms of genus 3 having a single zero. These curves were discovered by McMullen. The main invariants of our classification is…
We give an example of a Teichm\"uller curve which contains, in a factor of its monodromy, a group which was not observed before. Namely, it has Zariski closure equal to the group $SO^*(6)$ in its standard representation; up to finite index,…
Take a Teichm\"uller disc whose corresponding flat surface has a Veech-Group that contains a parabolic element. We look at its image in Schottkyspace and show that we can always construct a Schottky covering such that this image is not a…
We compute the rational cohomology of the moduli space of non-singular non-hyperelliptic complex projective curves of genus 3 with an odd theta characteristic.
We obtain the first complete classification result for algebraically primitive Teichm\"{u}ller curves in a stratum component in genus greater than 2. Specifically, we show that the Veech 14-gon generates the unique algebraically primitive…
Prym-Teichm\"uller curves $W_D(4)$ constitute the main examples of known primitive Teichm\"uller curves in the moduli space $\mathcal{M}_3$. We determine, for each non-square discriminant $D>1$, the number and type of orbifold points in…
The punctured solenoid $\S$ is an initial object for the category of punctured surfaces with morphisms given by finite covers branched only over the punctures. The (decorated) Teichm\"uller space of $\S$ is introduced, studied, and found to…
We study infinite superelliptic curves as translation surfaces and explore their Veech groups. These objects are branched covering of the complex plane with branching over infinitely many points. We provide a criterion for isomorphism…
We characterize subgroups of the mapping class group that stabilize a Teichmueller disk in terms of ellipses and strips that are immersed in the associated translation surface. In particular, we show that the space of immersed…
This work is motivated by two central questions in the birational geometry of moduli spaces of curves -- Fulton's conjecture and the effective cone of $\bar M_g$. We study the algebro-geometric aspect of Teichmuller curves parameterizing…
Quantum Teichmuller theory assigns invariants to three-manifolds via projective representations of mapping class groups derived from the representation of a noncommutative torus. Here, we focus on a representation of the simplest…
We study genus 3 hyperelliptic curves which have an extra involution. The locus $\L_3$ of these curves is a 3-dimensional subvariety in the genus 3 hyperelliptic moduli $\H_3$. We find a birational parametrization of this locus by affine…
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 these special Teichmuller discs is…
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…
We prove that there are no Shimura-Teichm\"uller curves generated by genus five translation surfaces, thereby completing the classification of Shimura-Teichm\"uller curves in general. This was conjectured by M\"oller in his original work…
In this work we study the additive orbifold cohomology of the moduli stack of smooth genus g curves. We show that this problem reduces to investigating the rational cohomology of moduli spaces of cyclic covers of curves where the genus of…
Teichmueller curves are geodesic discs in Teichmueller space that project to algebraic curves $C$ in the moduli space $M_g$. Some Teichm\"uller curves can be considered as components of Hurwitz spaces. We show that the absolute Galois group…
Triangulated surfaces are compact Riemann surfaces equipped with a conformal triangulation by equilateral triangles. In 2004, Brooks and Makover asked how triangulated surfaces are distributed in the moduli space of Riemann surfaces as the…
The Teichm\"{u}ller curve is the fiber space over Teichm\"{u}ller space of closed Riemann surfaces, where the fiber over a point in Teichm\"{u}ller space is the underlying surface. We derive formulas for sectional curvatures on the…