Lyapunov spectrum of square-tiled cyclic covers

无限族非循环的1-圆柱枕套铺砌曲面

An infinite family of non-cyclic 1-cylinder pillowcase-tiled surfaces

本文构造了无限族反例，反驳了Apisa-Wright关于所有在每个方向上至多有一个圆柱的二次微分分支覆盖都是循环覆盖的猜想。

Schwarz triangle mappings and Teichmüller curves: abelian square-tiled surfaces

We consider normal covers of CP^1 with abelian deck group, branched over at most four points. Families of such covers yield arithmetic Teichmüller curves, whose period mapping may be described geometrically in terms of Schwarz triangle mappings. These Teichmüller curves are generated by abelian square-tiled surfaces. We compute all individual Lyapunov exponents for abelian square-tiled surfaces, and demonstrate a direct and transparent dependence on the geometry of the period mapping. For this we develop a result of independent interest, which, for certain rank two bundles, expresses Lyapunov …

A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces

We present a Galois-theoretical criterion for the simplicity of the Lyapunov spectrum of the Kontsevich-Zorich cocycle over the Teichmueller flow on the $SL_2(R)$-orbit of a square-tiled surface. The simplicity of the Lyapunov spectrum has been proved by A. Avila and M. Viana with respect to the so-called Masur-Veech measures associated to connected components of moduli spaces of translation surfaces, but is not always true for square-tiled surfaces of genus $\geq 3$. We apply our criterion to square-tiled surfaces of genus 3 with one single zero. Conditionally to a conjecture of Delecroix and…

Square-tiled cyclic covers

A cyclic cover of the complex projective line branched at four appropriate points has a natural structure of a square-tiled surface. We describe the combinatorics of such a square-tiled surface, the geometry of the corresponding Teichmüller curve, and compute the Lyapunov exponents of the determinant bundle over the Teichmüller curve with respect to the geodesic flow. This paper includes a new example (announced by G. Forni and C. Matheus in \cite{Forni:Matheus}) of a Teichmüller curve of a square-tiled cyclic cover in a stratum of Abelian differentials in genus four with a maximally degenerat…

Square-tiled surfaces and rigid curves on moduli spaces

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 square-tiled surfaces with two applications: (a) there exist infinitely many rigid curves on the moduli space of hyperelliptic curves, they span the same extremal ray of the cone of moving curves and their union is Zariski dense, hence they yield infinitely many rigid curves with the same properties on the moduli space of stable n-pointed rational curves for e…

Un contre-exemple à la réciproque du critère de Forni pour la positivité des exposants de Lyapunov du cocycle de Kontsevich-Zorich

We introduce two square-tiled surfaces, one with $8$ squares inside $Ω\mathcal{M}_3(2,2)$, and the other with $9$ squares inside $Ω\mathcal{M}_4(3,3)$, respectively. In these examples, the dimensions of the isotropic subspaces (in absolute homology) generated by the waist curves of the maximal cylinders in any fixed rational direction are $2$ and $3$ respectively. Hence, a geometrical criterion of G. Forni for the non-uniform hyperbolicity of Kontsevich-Zorich (KZ) cocycle can not be applied to these examples. Nevertheless, we prove that there are no vanishing exponents and the spectrum is sim…