An algorithm for finding the Veech group of an origami

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.Comment: 17 pages, 1 figur

Lyapunov Exponents of Rank 2-Variations of Hodge Structures and Modular Embeddings

If the monodromy representation of a VHS over a hyperbolic curve stabilizes a rank two subspace, there is a single non-negative Lyapunov exponent associated with it. We derive an explicit formula using only the representation in the case when the monodromy is discrete.Comment: 22 pages, 4 figures; accepted version to be published in Ann. Inst. Fourier (Grenoble

General origamis and Veech groups of flat surfaces

In this century, a square-tiled translation surface (an origami) is intensively studied as an object with special properties of its translation structure and its $SL(2,\mathbb{R})$-orbit embedded in the moduli space. We generalize this concept in the language of flat surfaces appearing naturally in the Teichm\"uller theory. We study the combinatorial structure of origamis and show that a certain system of linear equations realizes the flat surface in which rectangles of specified moduli replace squares of an origami. This construction gives a parametrization of the family of flat surfaces with two finite Jenkins-Strebel directions for each combinatorial structure of two-directional cylinder decomposition. Moreover, we obtain the inclusion of Veech groups of such flat surfaces under a covering relation with specific branching behavior.Comment: 19 pages, 8 figure

Systolic geometry of translation surfaces

Let $S$ be a translation surface of genus $g > 1$ with $n$ cone points $(p_i)_{i=1,\ldots,n}$ with cone angle $2\pi \cdot (k_i+1)$ at $p_i$, where $k_i \in \mathbb{N}$. In this paper we investigate the systolic landscape of these translation surfaces for fixed genus.Comment: 25 pages, 4 figures. Added explicit computations of systoles in the graph of saddle connections for origamis in H(1,1) and a criterion to decide whether such systoles define systoles on the translation surfac