Congruence Veech Groups

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 translation surface which we call property $(\star)$. It guarantees that partition stabilising congruence subgroups of this level occur as Veech group of a translation covering. Each primitive surface with exactly one singular point has property $(\star)$ in every level. We additionally show that the surface glued from a regular $2n$-gon with odd $n$ has property $(\star)$ in level $a$ iff $a$ and $n$ are coprime. For the primitive translation surface glued from two regular $n$-gons, where $n$ is an odd number, we introduce a generalised Wohlfahrt level of subgroups in its Veech group. We determine the relationship between this Wohlfahrt level and the congruence level of a congruence group

Cylinder deformations in orbit closures of translation surfaces

Let M be a translation surface. We show that certain deformations of M supported on the set of all cylinders in a given direction remain in the GL(2,R)-orbit closure of M. Applications are given concerning complete periodicity, field of definition, and the number of of parallel cylinders which may be found on a translation surface in a given orbit closure. The proof uses Eskin-Mirzakhani-Mohammadi's recent theorem on orbit closures of translation surfaces, as well as results of Minsky-Weiss and Smillie-Weiss on the dynamics of horocycle flow.Comment: v2: Minor revision. Comments welcome! 24 page

Diophantine approximation on Veech surfaces

We show that Y. Cheung's general $Z$-continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates. The saddle connection continued fractions then allow one to recognize certain transcendental directions by their developments

On the Ergodicity of Flat Surfaces of Finite Area

We prove some ergodic theorems for flat surfaces of finite area. The first result concerns such surfaces whose Teichmuller orbits are recurrent to a compact subset of $SL(2;R)/SL(S)$, where $SL(S)$ is the Veech group of the surface. In this setting, this means that the translation flow on a flat surface can be renormalized through its Veech group to reveal ergodic properties of the translation flow. This result applies in particular to flat surfaces of infinite genus and finite area. Our second result is an criterion for ergodicity based on the control of deforming metric of a flat surface. Applied to translation flows on compact surfaces, it improves and generalizes a theorem of Cheung and Eskin.Comment: 23 pages. Accepted version to appear in GAF

Ergodicity for Infinite Periodic Translation Surfaces

For a Z-cover of a translation surface, which is a lattice surface, and which admits infinite strips, we prove that almost every direction for the straightline flow is ergodic