Wild translation surfaces and infinite genus

Abstract

The Gauß-Bonnet formula for classical translation surfaces relates the cone angle of the singularities (geometry) to the genus of the surface (topology). When considering more general translation surfaces, we observe so-called wild singularities for which the notion of cone angle is not applicable any more.In this article, we study whether there still exist relations between the geometry and the topology for translation surfaces with wild singularities. By considering short saddle connections, we determine under which conditions a wild singularity implies infinite genus. We apply this to show that the existence of a wild singularity on a parabolic or essentially finite translation surface implies infinite genus.Classical translation surfaces are objects at the intersection of many different fields such as dynamical systems, Teichmüller theory, algebraic geometry, topology, and geometric group theory. The history of translation surfaces starts in the time of the article [FK36]. Fox and Kershner obtained translation surfaces in the theory of billiards when "unfolding" polygons with rational angles.The most visual description of classical translation surfaces is given by considering finitely many polygons in the plane. If every edge of the polygons can be identified with a parallel edge of the same length so that we obtain a connected, orientable surface then the resulting object is a translation surface. It is locally flat at all points with the possible exception of the former vertices of the polygons. These exceptional points are called singularities and they are cone points of the resulting surface with cone angle 2 for some ≥ 2.A natural generalization is to drop the condition that the number of polygons has to be finite. When gluing infinitely many polygons, the local flatness still holds but the behaviour of the singularities is more diverse than in the classical case. This kind of translation surface is often called infinite in the literature, but we will not follow this convention here and simply call it translation surface. 1 arXiv:1410.1501v2 [math.GT] 18 Jan 2017 Recently, the interest in this generalization of translation surfaces has grown: There are results on Veech groups in [Cha04], [HS10], and [PSV11], results on the dynamics in [Hoo14], [Tre14], and [LT16], and results on infinite coverings of finite translation surfaces (especially for the wind-tree model) in [DHL14], [HLT11], [AHar], [HW12], [HHW13], and [FU14]. However, while we have a classification of finite translation surfaces by studying strata of the moduli space, there is no systematic description for the generalized ones so far. A first step towards such a classification can be to understand and classify the singularities of the translation surfaces. When considering translation surfaces with interesting singularities, it is natural to start with translation surfaces with exactly one singularity. So, a lot of the recently described examples are Loch Ness monsters, i.e. surfaces with infinite genus and one end (cf. [Ric63] for the defini...

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