arXiv research

Study covers of Chamanara surface with large Veech groups.

Arithmetic Kontsevich-Zorich monodromy found in a specific origami surface.

Study Veech groups in fibered 3-manifolds, proving no parabolics for fibers.

In this paper, we construct new examples of Veech groups by extending Schmithusen's method for calculating Veech groups of origamis to Veech groups of unramified finite coverings of regular 2n-gons. We calculate the Veech groups of certain Abelian coverings of regular 2n-gons by using an algebraic method.

Algorithm constructs surfaces with specific Veech groups in lattice strata.

We study the action of the Veech group of square-tiled surfaces of genus two on homology. This action defines the homology Veech group which is a subgroup of $\textrm{SL}_2(\mathcal{O}_D)$ where $\mathcal{O}_D$ is a quadratic order of square discriminant. Extending a result of Weitze-Schmithüsen we show that also the h…

Talked about the Veech group of a specific surface.

Origamis with specific groups have Veech groups that surject onto SL(2, Z/nZ).

A nice trick for studying the billiard flow in a rational polygon is to unfold the polygon along the trajectories. This gives rise to a translation or half-translation surface tiled by the original polygon, or equivalently an Abelian or quadratic differential. Veech surfaces are a special class of translation surfaces …

As main result we show that for each g > 1 there is some translation surface of genus g whose Veech group is a non congruence subgroup of SL(2,Z). We use origamis/square-tiled surfaces to produce our examples. The article is divided into two parts: In the first part we introduce translation surfaces, origamis, Veech gr…

We classify Veech groups of tame non-compact flat surfaces. In particular we prove that all countable subgroups of $\mathbf{GL}_+(2,\R)$ avoiding the set of mappings of norm less than 1 appear as Veech groups of tame non-compact flat surfaces which are Loch Ness monsters. Conversely, a Veech group of any tame flat surf…

The paper calculates Veech groups for triangulable structures on the sphere.

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 surfaces their Veech groups are subgroups of finite index of SL(2, Z). We show that each stratum of the space of translation sur…

Study infinite superelliptic curves and their Veech groups, providing geometric and algebraic insights.

We define an infinite series of translation coverings of Veech's double-n-gon for odd n greater or equal to 5 which share the same Veech group. Additionally we give an infinite series of translation coverings with constant Veech group of a regular n-gon for even n greater or equal to 8. These families give rise to expl…

Schmithüsen 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 framework of approachable Veech groups is greatly extended. In this paper, we continue the analysis and consider what kind of…

New groups are hyperbolic and rigid in mapping class groups.

We prove that there are finite area flat surfaces whose Veech group is an infinite cyclic group consisting of hyperbolic elements

We prove the existence of Veech groups having a critical exponent strictly greater than any elementary Fuchsian group (i.e. $>\frac{1}{2}$) but strictly smaller than any lattice (i.e. $<1$). More precisely, every affine covering of a primitive L-shaped Veech surface $X$ ramified over the singularity and a non-periodic …

We prove that a ``bouillabaisse'' surface (translation surface which has two transverse parabolic elements) has totally real trace field. As a corollary, non trivial Veech groups which have no parabolic elements do exist. The proof follows Veech's viewpoint on Thurston's construction of pseudo-Anosov diffeomorphisms.

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.

Veech groups uniformize Teichmüller geodesic curves in Riemann moduli space. Recently, examples of infinitely generated Veech groups have been given. We show that these can even have infinitely many cusps and infinitely many infinite ends. We further show that examples exist for which each direction of an infinite end …

Algorithm finds periodic points on Veech surfaces.

Extensions of Veech groups using hierarchical hyperbolic spaces.

The paper studies a group action on a hyperbolic space derived from a lattice Veech group.

We prove that every finite subgroup of $GL_{2}(\mathbb{R})$ can be realized as the Veech group of some translation surface.

We state conjectures on the asymptotic behavior of the Masur-Veech volumes of strata in the moduli spaces of meromorphic quadratic differentials and on the asymptotics of their area Siegel-Veech constants as the genus tends to infinity.

We show that every countable subgroup $G<\rm GL_+(2,\mathbb{R})$ without contracting elements is the Veech group of a tame translation surface $S$ of infinite genus, for infinitely many different topological types of $S$. Moreover, we prove that as long as every end has genus, there are no restrictions on the topologic…

We study Veech groups of covering surfaces of primitive translation surfaces. Therefore we define congruence subgroups in Veech groups of primitive translation surfaces using their action on the homology with entries in $\mathbb{Z}/a\mathbb{Z}$. We introduce a congruence level definition and a property of a primitive t…

Permutation of Weierstrass points on Veech surfaces in $\mathcal{H}(2)$ classified by discriminant.

We introduce a class of objects which we call 'affine surfaces'. These provide families of foliations on surfaces whose dynamics we are interested in. We present and analyze a couple of examples, and we define concepts related to these in order to motivate several questions and open problems. In particular we generalis…

Periodic points are points on Veech surfaces, whose orbit under the group of affine diffeomorphisms is finite. We characterise those points as being torsion points if the Veech surfaces is suitably mapped to its Jacobian or an appropriate factor thereof. For a primitive Veech surface in genus two we show that the only …

We extend asymptotic formulas for saddle connections on translation surfaces.

Flat surfaces that correspond to meromorphic $1$-forms or to meromorphic quadratic differentials containing poles of order two and higher are surfaces of infinite area. We classify groups that appear as Veech groups of translation surfaces with poles. We characterize those surfaces such that their $GL^{+}(2,\mathbb{R})…

These notes discuss an infinite translation surface, introduced by Chamanara. We review his proof that the Veech group is a non-elementary Fuchsian group of the second kind which is generated by two parabolic elements.

Formulae for Masur-Veech volumes and frequencies of geodesics derived from intersection numbers.

We describe Veech groups of flat surfaces arising from irrational angled polygonal billiards or irreducible stable abelian differentials. For irrational polygonal billiards, we prove that these groups are non-discrete subgroups of SO(2,R) and we calculate their rank.

We show that the Masur-Veech volumes and area Siegel-Veech constants can be obtained by intersection numbers on the strata of Abelian differentials with prescribed orders of zeros. As applications, we evaluate their large genus limits and compute the saddle connection Siegel-Veech constants for all strata. We also show…

New algorithm for computing Veech groups from translation surfaces.

We show that the triangle with angles Pi/12, Pi/3 and 7*Pi/12 has the lattice property and compute this triangle's Veech group.

The paper calculates Veech groups and Galois invariants for general origamis.

The paper studies the geometry and topology of a specific foliation on a complex surface.

In this paper, we study the PSV construction, which provides a step by step method for obtaining tame translation surfaces with a suitable Veech group. In addition, we modify slightly this construction, and for each finitely generated subgroup $G<{\rm GL}_{+}(2,\mathbb{R})$ without contracting elements, we produce a ta…

We compute explicitly the absolute contribution of square-tiled surfaces having a single horizontal cylinder to the Masur-Veech volume of any ambient stratum of Abelian differentials. The resulting count is particularly simple and efficient in the large genus asymptotics. Using the recent results of Aggarwal and of Che…

In this paper we consider the large genus asymptotics for two classes of Siegel-Veech constants associated with an arbitrary connected stratum $\mathcal{H} (α)$ of Abelian differentials. The first is the saddle connection Siegel-Veech constant $c_{\text{sc}}^{m_i, m_j} \big( \mathcal{H} (α) \big)$ counting saddle conne…

Quasimodular forms were first studied in the context of counting torus coverings. Here we show that a weighted version of these coverings with Siegel-Veech weights also provides quasimodular forms. We apply this to prove conjectures of Eskin and Zorich on the large genus limits of Masur-Veech volumes and of Siegel-Veec…

For every half-translation surface with marked points $(M,Σ)$, we construct an associated tessellation $Π(M,Σ)$ of the Poincaré upper half plane whose tiles have finitely many sides and area at most $π$. The tessellation $Π(M,Σ)$ is equivariant with respect to the action of $\mathrm{PSL}(2,\mathbb{R})$, and invariant w…

New loxodromic elements found in infinite-type surfaces.