A pseudo-trigonometry related to Ptolemy's theorem and the hyperbolic geometry of punctured spheres

• Autorzy: Joachim A. Hempel
• Języki publikacji: EN

Abstrakty

EN

A hyperbolic geodesic joining two punctures on a Riemann surface has infinite length. To obtain a useful distance-like quantity we define a finite pseudo-length of such a geodesic in terms of the hyperbolic length of its surrounding geodesic loop. There is a well defined angle between two geodesics meeting at a puncture, and our pseudo-trigonometry connects these angles with pseudo-lengths. We state and prove a theorem resembling Ptolemy's classical theorem on cyclic quadrilaterals and three general lemmas on intersections of shortest (in the sense of pseudo-length) geodesic joins. These ideas are then applied to the description of an optimal fundamental region for the covering Fuchsian group of a five-punctured sphere, effectively also giving a fundamental region for the modular group M(0,5).

Kategorie tematyczne

• 32G15: Moduli of Riemann surfaces, Teichm\"uller theory
• 30F35: Fuchsian groups and automorphic functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2004
• Tom: 84
• Numer: 2
• Strony: 147-167

Daty

wydano

2004

Twórcy

autor

Joachim A. Hempel

• School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, Australia

Identyfikatory

DOI

10.4064/ap84-2-5