zbMATH — the first resource for mathematics

Geometric structures on toroidal maps and elliptic curves. (English) Zbl 0984.05028
With every map on a surface there is an associated geometric structure (spherical, Euclidean, or hyperbolic) on the surface. A surface of genus $$>1$$ always carries a hyperbolic structure, but the structure on a torus can be either Euclidean or hyperbolic. The authors study the toroidal maps which have a Euclidean structure, both from the viewpoint of graph embeddings and of elliptic curves.

MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 57M15 Relations of low-dimensional topology with graph theory
Full Text:
References:
 [1] BELYI G. V.: On Galois extensions of a maximal cyclotomic field. Izv. Akad. Nauk SSSR 43 (1979), 269-276 (Russian) [ · Zbl 0409.12012 [2] BEAZLEYCOHEN P.-ITZYKSON C.-WOLFART J.: Fuchsian triangle groups and Grothendieck Dessins. Variations on a theme of Belyi. Comm. Math. Phys. 163 (1994), 605-627. · Zbl 0811.14030 [3] COXETER H. S. M.-MOSER W. O. J.: Generators and Relations for Discrete Groups. Springer-Verlag, Berlin, 1965. · Zbl 0133.28002 [4] GRIFFITHS P. A.: Introduction to Algebraic Curves. (2nd. Transl. Math. Monographs 76, Amer. Math. Soc, Providence, RI, 1996 · Zbl 0873.14030 [5] GROTHENDIECK A.: Esquisse d’un programme. Geometric Galois Actions 1 (L. Schneps, P. Lochak, London Math. Soc. Lecture Note Ser. 242, Cambridge University Press, Cambridge, 1997. · Zbl 0901.14001 [6] JONES G. A.-SINGERMAN D.: Theory of maps on orientable surfaces. Proc. London Math. Soc. (3) 37 (1978), 273-307. · Zbl 0391.05024 [7] JONES G. A.-SINGERMAN D.: Complex Functions, an Algebraic and Geometric Viewpoint. Cambridge University Press, Cambridge, 1987. · Zbl 0608.30001 [8] JONES G. A.-SINGERMAN D.: Belyi functions, hypermaps and Galois groups. Bull. London Math. Soc. 28 (1996), 561-590. · Zbl 0853.14017 [9] KNAPP A. W.: Elliptic curves. Math. Notes 40, Princeton Univ.Press, Princeton, NJ, 1992. · Zbl 0804.14013 [10] MAGNUS W.: Noneuclidean Tesselations and Their Groups. Academic Press, New York, 1974. · Zbl 0293.50002 [11] SINGERMAN D.-SYDDALL R. I.: Belyi uniformization of elliptic curves. Bull. London Math. Soc. 29 (1997), 443-451. · Zbl 0868.14019 [12] STARK H. M.: A complete determination of the complex quadratic fields of class number one. Michigan Math. J. 14 (1967), 1-27. · Zbl 0148.27802 [13] SYDDALL R. I.: Uniform maps on the torus. European J. Combin. · Zbl 1064.14030 [14] WOLFART J.: The obvious’ part of Belyi’s Theorem and Riemann surfaces with many automorphisms. Geometric Galois Actions 1 (L. Schneps, P. Lochak, London Math. Soc. Lecture Note Ser. 242, Cambridge University Press, Cambridge, 1997. · Zbl 0915.14021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.