Notes on the Veech group of the Chamanara surface

Abstract

These notes follow and extend the proof of the description of the Veech group in [Cha04, Theorem 4]. The main result is that the Veech group of the Chamanara surface is a non-elementary Fuchsian group of the second kind which is generated by two parabolic elements. Most of the calculations were carried out on the train from Marseille to Karlsruhe after the wonderful conference "Dynamics and Geometry in the Teichmüller Space" in July 2015.

Cyclic hyperbolic Veech groups in finite area

“…It follows for example from [2,3] (or also from [1]) that the group G of affine transformations of the Chamanara surface X is countable. It thus follows from Lemma 3.1 that the set F = {ζ ∈ X| Stab G (ζ) = Id} consists of countably many points and countably many straight segments.…”

“…The Chamanara surface. In this section we recall the construction of the Chamanara surface X (see [2] and [3] for details). We start with the closed unit square [0, 1] × [0, 1] in R 2 and consider for k = 0, 1, 2, .…”

Rotated Odometers

“…We show in Theorem 1.1 that such systems can be considered as first return maps of flows of rational slope on certain translation surfaces of finite area and infinite genus with finite number of ends. Flows of rational slope on translation surfaces can also be studied by very different geometrical methods, as for instance in [7,13], but nothing is known about flows of irrational slope. Therefore, the following question is natural.…”