Geometry of Infinitely Generated Veech Groups

O ct 2 00 4 GEOMETRY OF INFINITELY GENERATED VEECH GROUPS

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 is the limit of directions of inequivalent infinite ends.

Infinitely Generated Veech Groups

We give a positive response to a question of W. Veech: Infinitely generated Veech groups do exist.

Infinite Translation Surfaces with Infinitely Generated Veech Groups

Generalized Staircases: Recurrence and Symmetry

We study infinite translation surfaces which are Z-covers of compact translation surfaces. We obtain conditions ensuring that such surfaces have Veech groups which are Fuchsian of the first kind and give a necessary and sufficient condition for recurrence of their straight-line flows. Extending results of Hubert and Schmithüsen, we provide examples of infinite non-arithmetic lattice surfaces, a...

Veech Groups and Polygonal Coverings

We discuss branch points of a ne coverings and their e ects on Veech groups. In particular, this allows us to show that even if one polygon tiles another, the respective Veech groups are not necessarily commensurable. We also show that there is no universal bound on the number of Teichm uller disks passing through the same point of Teichm uller space and having incommensurable lattice Veech g...