172 research outputs found
Geometric structures on toroidal maps and elliptic curves
No full textFrom the work [JONES,~G.~A.---SINGERMAN,~D.: {\it Theory of maps on orientable surfaces\/}, Proc. London Math. Soc.~(3) {\bf 37} (1978), 273--307] or [GROTHENDIECK,~A.: {\it Esquisse d'un programme\/}. In: Geometric Galois Actions~1 (L.~Schneps, P.~Lochakeds, eds.). London Math. Soc. Lecture Note Ser.~242, Cambridge University Press, Cambridge, 1997] there is associated with every map on a surface, a geometric structure on the surface, which is either spherical, Euclidean, or hyperbolic. A surface of genus~1 necessarily has a hyperbolic structure, but the torus can have either a Euclidean or hyperbolic structure. We study the genus~1 maps which have a Euclidean structure, both from the viewpoint of graph embeddings and of elliptic curves. We also find an embedding of the complete graph K_6 which necessarily has a hyperbolic structure and where the edges are hyperbolic geodesics
Real Belyi theory
No full textWe develop a Belyi type theory that applies to Klein surfaces, i.e. (possibly non-orientable) surfaces with boundary which carry a dianalytic structure. In particular we extend Belyi's famous theorem from Riemann surfaces to Klein surfaces
Superficies de riemann y cristalografia
No full textThis article explains some connections between Euclidean crystallogaphic groups and complex structures on the torus
Belyi uniformization of elliptic curves
No full textBelyi's Theorem implies that a Riemann surface X represents a curve defined over a number field if and only if it can be expressed as U/?, where U is simply-connected and ? is a subgroup of finite index in a triangle group. We consider the case when X has genus 1, and ask for which curves and number fields ? can be chosen to be a lattice. As an application, we give examples of Galois actions on Grothendieck dessins
On the fixed-point set of automorphisms of non-orientable surfaces without boundary
No full textMacbeath gave a formula for the number of fixed points for each non-identity element of a cyclic group of automorphisms of a compact Riemann surface in terms of the universal covering transformation group of the cyclic group. We observe that this formula generalizes to determine the fixed-point set of each non-identity element of a cyclic group of automorphisms acting on a closed non-orientable surface with one exception; namely, when this element has order 2. In this case the fixed-point set may have simple closed curves (called ovals) as well as fixed points. In this note we extend Macbeath’s results to include the number of ovals and also determine whether they are twisted or not
Riemann surfaces, Belyi functions and hypermaps
No full textWe give some consequences of Belyi's Theorem for Riemann surface theory
Children's Citrus Activity: Citrus Counting
No full textFlorida is well known for its citrus industry, valued at over eight billion dollars, and is one of the top citrus-producing states in the United States. This new one-page children’s activity sheet about Florida citrus includes an activity for students learning to count and match. Written by Jamie D. Burrow and Ariel Singerman and published by the UF/IFAS Extension 4-H Youth Development Program.
https://edis.ifas.ufl.edu/4h40
A note on Belyi's theorem for Klein surfaces
No full textSingerman and the first named author have recently developed a real Belyi thoery, leaving open a particular case in the proof of Belyi's theorem for Klein surfaces. We answer their question affirmatively by a descent argument which turns out to extend to a much more general context
Regular maps and principal congruence subgroups of Hecke groups
No full textAbstractRegular q-valent maps correspond to normal subgroups of the triangle group (2,q,∞). This group has a representation as the Hecke group Hq which is generated by z→−1z and z→−1z+λq, where λq≔2cosπq. We investigate the regular maps corresponding to the principal congruence subgroups of Hq. Those of low index give many interesting regular maps
-gonal tessellations and their Petrie paths
No full textIn [Sin88] the second author showed that the Farey map F is a model of the universal triangular tessellation. This is a tessellation that covers every other triangular tessellation on an orientable surface. More precisely, the automorphism group of F is the classical modular group Γ = PSL(2, Z) and every triangular map is the quotient of F by a subgroup of Γ. The aim of this paper is to describe universal q-gonal tessellations. Here the modular group will be replaced by Hecke groups. In [SS16] it was shown that the Petrie paths of the Farey map pass through vertices whose numerators and denominators are Fibonacci numbers. In section 8 we consider Hecke-Fibonacci sequences which arise out of universal q-gonal tessellations.</p
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