1,720,982 research outputs found
Unicellular dessins and a uniqueness theorem for Klein's Riemann surface of genus 3
No full textIf we consider the 14-sided hyperbolic polygon of Felix Klein that defines his famous surface of genus 3, we have a unifacial dessin whose automorphism group is transitive on the edges but not on the directed edges of the dessin. We show that Klein's surface is the unique platonic surface with this property
Real Belyi theory
Full text linkWe 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
Non-maximal cyclic group actions on compact Riemann surfaces
Full text linkWe say that a finite group G of automorphisms of a Riemann surface X is non-maximal in genus y if (i) G acta as a group of automorphisms of some compact Riemann surface Xg of genus g and (ii), for alí such surfaces Xg, | Aut Xg |>| G |. In this paper we investigate ihe case where G is a cylic group Cn of order n. If Cn acts on only finitely many surfaces of genus g, then we completely solve the problem of finding all such pairs (n, g)
Geometric structures on toroidal maps and elliptic curves
Full text linkFrom 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
The Riemann Surface of a Uniform Dessin
No full textWe say that a Dessin (or hypermap) is uniform if all vertices, (resp. edges, faces) have the same valency. Associated to a dessin there is a well-defined compact Riemann surface, or complex algebraic curve, and in this work we make a study of the Riemann surfaces that arise from Uniform dessins
From Farey Fractions to the Klein Quartic and beyond
No full textIn his 1878/79 paper "Ueber die transformation siebenter ordnung der elliptischen functionen", 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 this idea from 7 to 11 as Klein attempted to do in his follow up paper "Ueber die transformation elfter ordnung der elliptischen functionen", in 1879
From Farey fractions to the Klein quartic and beyond
No full textIn 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 polygonfor Klein’s surface using arithmetic. Now the side pairings are immediate and essentiallythe same as in Klein’s paper. We also extend his work from 7 to 11 as Klein also did in afollow-up paper of 187
From Farey fractions to the Klein quartic and beyond
No full textIn 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
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
- …
