### Impacto

Etayo Gordejuela, J. Javier and Martínez, Ernesto
(1988)
*Hyperbolic polygons and NEC groups.*
Mathematical Proceedings of the Cambridge Philosophical Society, 104
.
pp. 261-272.
ISSN 0305-0041

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Official URL: http://journals.cambridge.org/abstract_S0305004100065439

## Abstract

Beardon gave a procedure for constructing a polygon with prescribed angles. For each ordered set of angles Beardon's polygon is unique up to congruence. The polygon obtained this way has an inscribed circle. It is possible to obtain by means of these polygons a fundamental region for a non-Euclidean crystallographic (NEC) group with a given signature having equal angles in each of the cycles. In [5] the minimal number of sides of a convex fundamental region of an NEC group is calculated, and regions are explicitly obtained achieving the bound.

Item Type: | Article |
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Uncontrolled Keywords: | Moduli of Riemann surfaces, Teichmüller theory; Structure of modular groups and generalizations; arithmetic groups; Riemann surfaces; Hyperbolic and elliptic geometries (general) and generalizations; Topology of E 2 , 2 -manifolds |

Subjects: | Sciences > Mathematics > Algebraic geometry |

ID Code: | 15768 |

References: | Beardon, A. F.. Hyperbolic polygons and Fuchsian groups. J. London Math. Soc. (2)20 (1979), 247–254. Beardon, A. F.. The Geometry of Discrete Groups. Graduate Texts in Math. no. 91 (Springer-Verlag, 1983). Macbeath, A. M.. The classification of non-Euclidean plane crystallographic groups. Canad. J. Math. 19 (1967), 1192–1205. Macbeath, A. M. and Singerman, D.. Spaces of subgroups and Teichmüller space. Proc. London Math. Soc. (3) 31(1975), 211–256. Martínez, E.. Convex fundamental regions for N.E.C. groups. Arch. Math. (Basel) 47 (1986), 457–464. Singerman, P.. Symmetries of Riemann surfaces with large automorphism group. Math. Ann. 210 (1974), 17–32. Singerman, D.. On the structure of non-Euclidean crystallographic groups. Proc. Cambridge Philos. Soc. 76 (1974), 233–240. Wilkie, H. C.. On non-Euclidean crystallographic groups. Math. Z. 91(1966), 87–102. Zieschang, H., Vogt, E. and Coldewey, H. D.. Surfaces and Planar Discontinuous Groups. Lecture Notes in Math. vol. 835 (Springer-Verlag, 1980). |

Deposited On: | 26 Jun 2012 11:19 |

Last Modified: | 06 Feb 2014 10:31 |

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