The GauΓ-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, TeichmΓΌller 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 β₯ 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...
We study an interval exchange transformation of [0, 1] formed by cutting the interval at the points 1 n and reversing the order of the intervals. We find that the transformation is periodic away from a Cantor set of Hausdorff dimension zero. On the Cantor set, the dynamics are nearly conjugate to the 2-adic odometer.
For a half-translation surface (π, π), the associated saddle connection complex π(π, π) is the simplicial complex where vertices are the saddle connections on (π, π), with simplices spanned by sets of pairwise disjoint saddle connections. This complex can be naturally regarded as an induced subcomplex of the arc complex. We prove that any simplicial isomorphism π : π(π, π) β π(π β² , π β² ) between saddle connection complexes is induced by an affine diffeomorphism πΉ : (π, π) β (π β² , π β² ). In particular, this shows that the saddle connection complex is a complete invariant of affine equivalence classes of half-translation surfaces. Throughout our proof, we develop several purely combinatorial criteria for detecting various geometric objects on a half-translation surface, which may be of independent interest.
We generalize the notion of cusp excursion of geodesic rays by introducing for any k β₯ 1 the k th excursion in the cusps of a hyperbolic N -manifold of finite volume. We show that on one hand, this excursion is at most linear for geodesics that are generic with respect to the hitting measure of a random walk. On the other hand, for k β₯ N β 1, the k th excursion is superlinear for geodesics that are generic with respect to the Lebesgue measure. We use this to show that the hitting measure and the Lebesgue measure on the boundary of hyperbolic space H N for any N β₯ 2 are mutually singular.
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 TeichmΓΌller Space" in July 2015.
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