Fundamental Polygon Research Articles

Diameter of Compact Riemann Surfaces

AbstractDiameter is one of the most basic properties of a geometric object, while Riemann surfaces are one of the most basic geometric objects. Surprisingly, the diameter of compact Riemann surfaces is known exactly only for the sphere and the torus. For higher genuses, only very general but loose upper and lower bounds are available. The problem of calculating the diameter exactly has been intractable since there is no simple expression for the distance between a pair of points on a high-genus surface. Here we prove that the diameters of a class of simple Riemann surfaces known as generalized Bolza surfaces of any genus greater than 1 are equal to the radii of their fundamental polygons. This is the first exact result for the diameter of a compact hyperbolic manifold.

Cutting the Möbius Strip

Summary Using tools from algebraic topology and algebraic geometry (fundamental polygons and blowups, respectively), we prove that cutting along the center of a Möbius strip yields a cylinder.

A few words about maps

In this paper, we survey some properties, encoding, and bijections involving combinatorial maps, double occurrence words, and chord diagrams. We particularly study quasi-trees from a purely combinatorial point of view and derive a topological representation of maps with a given spanning quasi-tree using two fundamental polygons, which extends the representation of planar maps based on the equivalence with bipartite circle graphs. Then, we focus on Depth-First Search trees and their connection with a poset we define on the spanning quasi-trees of a map. We apply the bijections obtained in the first section to the problem of enumerating loopless rooted maps. Finally, we return to the planar case and discuss a decomposition of planar rooted loopless maps and its consequences on planar rooted loopless map enumeration.

Rigidity and flexibility of entropies of boundary maps associated to Fuchsian groups

Given a closed, orientable surface of constant negative curvature and genus $g \ge 2$, we study the topological entropy and measure-theoretic entropy (with respect to a smooth invariant measure) of generalized Bowen--Series boundary maps. Each such map is defined for a particular fundamental polygon for the surface and a particular multi-parameter. We present and sketch the proofs of two strikingly different results: topological entropy is constant in this entire family ("rigidity"), while measure-theoretic entropy varies within Teichm\"uller space, taking all values ("flexibility") between zero and a maximum, which is achieved on the surface that admits a regular fundamental $(8g-4)$-gon. We obtain explicit formulas for both entropies. The rigidity proof uses conjugation to maps of constant slope, while the flexibility proof -- valid only for certain multi-parameters -- uses the realization of geodesic flow as a special flow over the natural extension of the boundary map.

From Farey fractions to the Klein quartic and beyond

In a paper published in 1878/79 Klein produced his famous 14 -sided polygon representing the Klein quartic, his Riemann surface of genus 3 which has PSL(2, 7) as its automorphism group. The construction and method of side pairings are fairly complicated. By considering the Farey map modulo 7 we show how to obtain a fundamental polygon for Klein’s surface using arithmetic. Now the side pairings are immediate and essentially the same as in Klein’s paper. We also extend his work from 7 to 11 as Klein also did in a follow-up paper of 1879.

Uniformization of hyperelliptic curves as a systematic approach to establishing decision regions of hyperbolic signal sets

In this paper, we present a new approach to designing hyperbolic signal sets matched to groups using Whittaker’s proposal in the uniformization of hyperelliptic curves via Fuchsian differential equations (FDEs). This approach provides a systematic procedure of establishing the decision region (fundamental polygon) of a hyperbolic signal as well as the associated Fuchsian group, and consists of the following four steps: (1) Obtaining the genus, g, by embedding a discrete memoryless channel (DMC) on a Riemann surface; (2) Selecting a set of $$(2g+1)$$ or $$(2g+2)$$ symmetric points in the Poincare disk to establish the hyperelliptic curve; (3) From the FDE via algebraic manipulations to arrive at the hypergeometric differential equation (HDE) to obtain the solutions; and (4) Quotients of the FDE linearly independent solutions give rise to the generators of the associated Fuchsian group whose subgroup provides the uniformizing region, implying the determination of the decision region (Voronoi region) of a digital signal. Hence, the following results are achieved: (1) from the solutions of the FDE, the Fuchsian group and subgroup generators are established, and consequently, the desired signal set matched to the Fuchsian subgroup may be constructed; (2) a relation between the parameters of the $$\{p,q\}$$ tessellation and the degree of the hyperelliptic curve is established. Knowing g, related to the hyperelliptic curve degree, and p, the number of sides of the fundamental polygon derived from Whittaker’s uniformizing procedure, the value of q is obtained from the Euler characteristic leading to one of the $$\{4g,4g\}$$ or $$\{4g+2, 2g+1\}$$ or $$\{12g-6,3\}$$ tessellation. These tessellations are essential for their rich geometric and algebraic structures, both required in classical and quantum coding theory applications.

Flexibility of measure-theoretic entropy of boundary maps associated to Fuchsian groups

AbstractGiven a closed, orientable, compact surface S of constant negative curvature and genus $g \geq 2$ , we study the measure-theoretic entropy of the Bowen–Series boundary map with respect to its smooth invariant measure. We obtain an explicit formula for the entropy that only depends on the perimeter of the $(8g-4)$ -sided fundamental polygon of the surface S and its genus. Using this, we analyze how the entropy changes in the Teichmüller space of S and prove the following flexibility result: the measure-theoretic entropy takes all values between 0 and a maximum that is achieved on the surface that admits a regular $(8g-4)$ -sided fundamental polygon. We also compare the measure-theoretic entropy to the topological entropy of these maps and show that the smooth invariant measure is not a measure of maximal entropy.

Extremal parameters and their duals for boundary maps associated to Fuchsian groups

We describe arithmetic cross-sections for geodesic flow on compact surfaces of constant negative curvature using generalized Bowen–Series boundary maps, and their natural extensions, associated to co-compact torsion-free Fuchsian groups. If the boundary map parameters are extremal—that is, each is an endpoint of a geodesic that extends a side of the fundamental polygon—then the natural extension map has a domain with finite rectangular structure, and the associated arithmetic cross-section is parametrized by this set. This construction allows us to represent the geodesic flow as a special flow over a symbolic system of coding sequences. Moreover, each extremal parameter choice has a corresponding dual parameter choice such that the “past” of the arithmetic code of a geodesic is the “future” for the code using the dual parameter. This duality was observed for two classical parameter choices by Adler and Flatto; here we show constructively that every extremal parameter choice has a dual.

A Topological Transformation of Quantum Dynamics

In this work, we discuss the topological transformation of quantum dynamics by showing the wave dynamics of a quantum particle on different types of topological structures in various dimensions from the fundamental polygons of the corresponding universal covering spaces. This is not the view from different perspectives of an observer who simply uses different coordinate systems to describe the same physical phenomenon but rather possible geometric and topological structures that quantum particles are endowed with when they are identified with differentiable manifolds that are embedded or immersed in Euclidean spaces of higher dimension. We present our discussions in the form of Bohr model in one, two and three dimensions using linear wave equations. In one dimension, the fundamental polygon is an interval and the universal covering space is the straight line and in this case the standing wave on a finite string is transformed into the standing wave on a circle which can be applied into the Bohr model of the hydrogen atom. In two dimensions, the fundamental polygon is a square and the universal covering space is the plane and in this case, the standing wave on the square is transformed into the standing wave on different surfaces that can be formed by gluing opposite sides of the square, which include a 2-sphere, a 2-torus, a Klein bottle and a projective plane. In three dimensions, the fundamental polygon is a cube and the universal covering space is the three-dimensional Euclidean space. It is shown that a 3-torus and the manifold K × S1 defined as the product of a Klein bottle and a circle can be constructed by gluing opposite faces of a cube. Therefore, in three-dimensions, the standing wave on a cube is transformed into the standing wave on a 3-torus or on the manifold K × S1. We also suggest that the mathematical degeneracy may play an important role in quantum dynamics and be associated with the concept of wavefunction collapse in quantum mechanics.

Computing Surface Uniformization Using Discrete Beltrami Flow

In this paper, we propose a novel algorithm for computing surface uniformization for surfaces with arbitrary topology. According to the celebrated uniformization theorem, all Riemann surfaces can be classified as elliptic, parabolic, or hyperbolic. Our algorithm is able to work on all these cases by first constructing an initial map onto an appropriate domain, such as a sphere, or a polygon in the plane $\mathbb{R}^2$ or the hyperbolic disk $D$, and then morphing the diffeomorphism based on the discrete Beltrami flow algorithm. For high genus surfaces, both the final mapping and the target domain are unknown, which presents a challenge in general. Each such surface can be conformally mapped onto $D$ modulo a discrete subgroup of all fractional linear transforms on $D$. A conformal copy of the surface, which is also its uniformization domain, can be visualized as a fundamental polygon in $D$ corresponding to the discrete subgroup, where the generators of the discrete subgroup map each side of the fundamental polygon to its corresponding side, giving the conformality information of the surface. The novelty in our method lies in the iterative change of these generators as the diffeomorphism is morphing, which indicates a change of geometry of the target domain to match the geometry of the original surface. Numerical results are presented to show the efficiency and accuracy (in terms of distortion) of our methods as well as to make a comparison to other state-of-the-art algorithms.

Geometry curves: A compact representation for 3D shapes

We propose a novel compact surface representation, namely geometry curves, which record the essence of shape geometry and topology. The geometry curves mainly contain two parts: the interior and boundary lines. The interior lines, which correspond to the feature lines, record the geometry information of the 3D shapes; the boundary lines, which correspond to the boundary or fundamental polygons, record the topology information of the 3D shapes. As a vector representation, geometry curves can depict highly complex geometry details. The concept of geometry curves can be utilized in many potential applications, e.g., mesh compression, shape modeling and editing, animation, and level of details. Furthermore, we develop a procedure for automatically constructing geometry curves which obtain an excellent approximation to the original mesh.

Generic fundamental polygons for Fuchsian groups

A Dirichlet fundamental polygon for a Fuchsian group is said to be generic if its combinatorial shape is stable under any small permutation of the center of the polygon. Almost all points in the hyperbolic plane are known to be centers of generic fundamental polygons. We prove that the same property holds for points in the boundary of the hyperbolic plane.

Geometrical study of prime order automorphisms on Klein surfaces

We consider (n,q)-gonal Klein surfaces of algebraic genus p≥2 for which n≥p and n prime. We exclude the case n=2 corresponding to q-hyperelliptic surfaces, so n≥3. We study geometrical conditions on fundamental polygons of the NEC groups which uniformize these surfaces. From these conditions we give a way to construct such surfaces and obtain parameters which characterize them.

From Farey symbols to generators for subgroups of finite index in integral group rings of finite groups

AbstractThe topic of this paper is the construction of a finite set of generators for a subgroup of finite index in the unit group u(ℤG) of the integral group ring of a finite group G. The present paper is a continuation of earlier research by Bass and Milnor, Jespers and Leal, and Ritter and Sehgal who constructed such generators provided that the group G does not have a non-abelian fixed-point free epimorphic image and the rational group algebra ℚG does not have simple epimorphic images that are two-by-two matrices over either the rationals, a quadratic imaginary extension of the rationals or a non-commutative division algebra. In this paper we allow simple images of the type M2(ℚ). We will do so by introducing new additional generators using Farey symbols, which are in one to one correspondence with fundamental polygons of congruence subgroups of PSL2(ℤ). Furthermore, for each simple Wedderburn component M2(ℚ) of ℚG, the new generators give a free subgroup that is embedded in M2(ℤ).

On Uniformization of Algebraic Curves

Based on Burnside's parametrization of the algebraic curve $y^2=x^5-x$ we provide remaining attributes of its uniformization: Fuchsian equations and their solutions, accessory parameters, monodromies, conformal maps, fundamental polygons, etc. As a generalization, we construct the zero genus uniformization of arbitrary curves. For hyperelliptic curves all the objects of the theory are explicitly described. We consider a large number of examples and, briefly, applications: Abelian integrals, metrics of Poincare, differential equations of the Jacobi--Chazy and Picard--Fuchs type, and others.

Substitutional structures in simply and multiconnected flat spaces, with astrophysical applications

Hexagonal, octagonal and dodecagonal tilings of both simply and multiconnected flat spaces in two dimensions are considered. The tessellations of the Euclidean plane have, in common, arrowed prototiles, which are used for the construction of fundamental polygons for the flat torus and the Klein bottle. Non-deterministic derivations in formal grammars, producing non-periodic ordered structures, have been introduced recently also for the analysis of variable stars with multiple periods, like the semiregular star UW Her and the δ-Scuti star V784 Cas. Observations of the V346 Ori light curve can be modelled with similar techniques.

Effective Solution of the Carleman Problem for Some Groups of Divergent Type

We consider a Fuchsian group of the first kind containing only hyperbolic transformations. We obtain an effective solution (explicit in terms of transformations of the group) to the Carleman boundary value problem on a fundamental polygon where the inverse shift is induced by the generating transformations of the group.

Extremal disks and extremal surfaces of genus three

A compact Riemann surface of genus g ≥ 2 is said to be extremal if it admits an extremal disk, a disk of the maximal radius determined by g. If g = 2 or g ≥ 4, it is known that how many extremal disks an extremal surface of genus g can admit. In the present paper we deal with the case of g = 3. Considering the side-pairing patterns of the fundamental polygons, we show that extremal surfaces of genus 3 admit at most two extremal disks and that 16 surfaces admit exactly two. Also we describe the group of automorphisms and hyperelliptic surfaces.

On Rademacher's conjecture: congruence subgroups of genus zero of the modular group

We list all genus zero congruence subgroups of PSL 2 ( Z ) . There are altogether 132 of them (up to conjugation in PSL 2 ( Z ) ). Geometrical invariants (genus, v 2 , v 3 , number of cusps, index in PSL 2 ( Z ) ), fundamental polygons, Farey symbol, and independent generators of all such groups are determined. Generators of the function fields associated to such groups are also determined ( http://www.math.nus.edu.sg/~matlml/ ).

On Rademacher's conjecture: congruence subgroups of genus zero of the modular group

We list all genus zero congruence subgroups of PSL 2( Z) . There are altogether 132 of them (up to conjugation in PSL 2( Z) ). Geometrical invariants (genus, v 2, v 3, number of cusps, index in PSL 2( Z) ), fundamental polygons, Farey symbol, and independent generators of all such groups are determined. Generators of the function fields associated to such groups are also determined ( http://www.math.nus.edu.sg/~matlml/).