Related papers: Systoles in translation surfaces
The Thurston spine consists of the subset of Teichm\"uller space at which the set of shortest curves, the systoles, cuts the surface into polygons. The systole function is a topological Morse function on Teichm\"uller space. This paper…
The complete classification of hexapods - also known as Stewart Gough platforms - of mobility one is still open. To tackle this problem, we can associate to each hexapod of mobility one an algebraic curve, called the configuration curve. In…
The main goal of this article is to understand how the length spectrum of a random surface depends on its genus. Here a random surface means a surface obtained by randomly gluing together an even number of triangles carrying a fixed metric.…
For hyperbolic surfaces with geodesic boundary, we study the orthosystole, i.e. the length of a shortest essential arc from the boundary to the boundary. We recover and extend work by Bavard completely characterizing the surfaces maximizing…
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…
We prove that the asymptotic number of pairs of saddle connections with length smaller than $L$ with bounded virtual area is quadratic for almost every translation surface with respect to any ergodic $SL(2,\mathbb{R})$-invariant measure. A…
Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surface…
A common measure of a function's complexity is the count of its stationary points. For complicated functions, this count grows exponentially with the volume and dimension of their domain. In practice, the count is averaged over a class of…
An estimate for the genus function in circle bundles over irreducible 3-manifolds is proven. This estimate is in many cases an equality and it relates the minimal genus of the surfaces representing a given homology class with the…
We consider translation surfaces with poles on surfaces. We shall prove that any finite group appears as the automorphism group of some translation surface with poles. As a direct consequence we obtain the existence of structures achieving…
We consider the ortho spectrum of hyperbolic surfaces with totally geodesic boundary. We show that in general the ortho spectrum does not determine the systolic length but that there are only finitely many possibilities. As a corollary we…
In this paper, we give the maximum of the numbers $n$ such that we can take $n$ simple closed geodesics without singularities that are disjoint to each other for translation surfaces in the hyperelliptic components $\mathcal{H}^{\rm…
We compute the number of systoles, the shortest simple closed geodesics and 2-systoles, the second shortest simple closed geodesics on hyperbolic surfaces homeomorphic to once-punctured torus and four-punctured sphere.
In this paper, we study and classify singular minimal translation surfaces in a Euclidean space of dimension 3 endowed with a certain semi-symmetric (non-)metric connection.
We consider how quickly a typical point returns to neighborhoods of itself under the flow in a typical direction on a translation surface.
The Nielsen-Thurston theory of surface diffeomorphisms shows that useful dynamical information can be obtained about a surface diffeomorphism from a finite collection of periodic orbits.In this paper, we extend these results to homoclinic…
This paper deals with iteration stable (STIT) tessellations, and, more generally, with a certain class of tessellations that are infinitely divisible with respect to iteration. They form a new, rich and flexible class of spatio-temporal…
A translation surface in Lorentz-Minkowski space $\rr^3$ is a surface defined as the sum of two spatial curves. In this paper we present a classification of maximal surfaces of translation type. We prove that if a generating curve is…
This book explores infinite-type translation surfaces and is intended as an introductory text for graduate and PhD students, as well as a reference for more advanced researchers. Chapter 1 introduces the three definitions of translation…
The so-called {\it kissing number} for hyperbolic surfaces is the maximum number of homotopically distinct systoles a surface of given genus $g$ can have. These numbers, first studied (and named) by Schmutz Schaller by analogy with lattice…