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Publication# Computational Approach to the Geometry of Compact Riemann Surfaces

Abstract

The goal of this document is to provide a generalmethod for the computational approach to the topology and geometry of compact Riemann surfaces. The approach is inspired by the paradigms of object oriented programming. Our methods allow us in particular to model, for numerical and computational purposes, a compact Riemann surface given by Fenchel-Nielsen parameters with respect to an arbitrary underlying graph, this in a uniformand robust manner. With this programming model established we proceed by proposing an algorithmthat produces explicit compact fundamental domains of compact Riemann surfaces as well as generators of the corresponding Fuchsian groups. In particular, we shall explain how onemay obtain convex geodesic canonical fundamental polygons. In a second part we explain in what manner simple closed geodesics are represented in our model. This will lead us to an algorithm that enumerates all these geodesics up to a given prescribed length. Finally, we shall briefly overview a number of possible applications of our method, such as finding the systoles of a Riemann surface, or drawing its Birman-Series set in a fundamental domain.

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Related concepts (12)

Related publications (3)

Geodesic

In geometry, a geodesic (ˌdʒiː.əˈdɛsɪk,*-oʊ-,*-ˈdiːsɪk,_-zɪk) is a curve representing in some sense the shortest path (arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line". The noun geodesic and the adjective geodetic come from geodesy, the science of measuring the size and shape of Earth, though many of the underlying principles can be applied to any ellipsoidal geometry.

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.

Length

Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the International System of Units (SI) system the base unit for length is the metre. Length is commonly understood to mean the most extended dimension of a fixed object. However, this is not always the case and may depend on the position the object is in.

To any compact Riemann surface of genus g one may assign a principally polarized abelian variety (PPAV) of dimension g, the Jacobian of the Riemann surface. The Jacobian is a complex torus and we call a Gram matrix of the lattice of a Jacobian a period Gram matrix. The aim of this thesis is to contribute to the Schottky problem, which is to discern the Jacobians among the PPAVs. Buser and Sarnak approached this problem by means of a geometric invariant, the first successive minimum. They showed that the square of the first successive minimum, the squared norm of the shortest non-zero vector, in the lattice of a Jacobian of a Riemann surface of genus g is bounded from above by log(4g), whereas it can be of order g for the lattice of a PPAV of dimension g. The main goal of this work was to improve this result and to get insight into the connection between the geometry of a compact Riemann surface that is given in hyperbolic geometric terms, and the geometry of its Jacobian. We show the following general findings: For a hyperelliptic surface the first successive minimum is bounded from above by a universal constant. The square of the second successive minimum of the Jacobian of a Riemann surface of genus g is equally of order log(g). We provide refined upper bounds on the consecutive successive minima if the surface contains several disjoint small simple closed geodesics and a lower bound for the norm of certain lattice vectors of the Jacobian, if the surface contains small non-separating simple closed geodesics. If the concrete geometry of the Riemann surface is known, more precise statements can be made. In this case we obtain theoretical and practical estimates on all entries of the period Gram matrix. Here we establish upper and lower bounds based on the geometry of the cut locus of simple closed geodesics and also on the geometry of Q-pieces. In addition the following two results have been obtained: First, an improved lower bound for the maximum value of the norm of the shortest non-zero lattice vector among all PPAVs in even dimensions. This follows from an averaging method from the geometry of numbers applied to a family of symmetric PPAVs. Second, a new proof for a lower bound on the number of homotopically distinct geodesic loops, whose length is smaller than a fixed constant. This lower bound applies not only to geodesic loops on Riemann surfaces, but on arbitrary manifolds of non-positive curvature.

Using an algebraic formalism based on matrices in SL(2,R), we explicitly give the Teichmüller spaces of Riemann surfaces of signature (0,4) (X pieces), (1,2) ("Fish" pieces) and (2,0) in trace coordinates. The approach, based upon gluing together two building blocks (Q and Y pieces), is then extended to tree-like pants decomposition for higher signatures (g,n) and limit cases such as surfaces with cusps or cone-like singularities. Given the Teichmüller spaces, we establish a set of generators of their modular groups for signatures (0,4), (1,2) and (2,0) in trace coordinates using transformations acting separately on the building blocks and an algorithm on dividing geodesics. The fact that these generators act particularly nice in trace coordinates gives further motivation to this choice (rather then the one of Fenchel-Nielsen coordinates). This allows us to solve the Riemann moduli problem for X pieces, "Fish" pieces and surfaces of genus 2; i.e. to give the moduli spaces as the fundamental domains for the action of the modular groups on the Teichmüller spaces. In this context, we also give an algorithm deciding whether two Riemann surfaces of signatures (0,4), (1,2) or (2,0) given by points in the Teichmüller space are isometric or not. As a consequence, we show the following two results concerning simple closed geodesics: On any purely hyperbolic Riemann surface (containing neither cusps nor cone-like singularities), the longest of two simple closed geodesics that intersect one another n times is of length at least ln, a sharp constant independent of the surface. We explicitly give ln for n = 1,2,3 and study its behaviour when n goes to infinity. X pieces are spectrally rigid with respect to the length spectrum of simple closed geodesics.

Klaus-Dieter Semmler, Thomas Gauglhofer

Summary: We explicitly give $calT$, the Teichmüller space of four-holed spheres (which we call $X$ pieces) in trace coordinates, as well as its modular group and a fundamental domain for the action of this group on $calT$ which is its moduli space. As a consequence, we see that on any hyperbolic Riemann surface, two closed geodesics of lengths smaller than $2operatornamearccosh(2)$ intersect at most once; two closed geodesics of lengths smaller than $2operatornamearccosh(3)$ are both non-dividing or intersect at most once.

2005