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

Joachim A. Hempel¹

¹ School of Mathematics and Statistics The University of Sydney Sydney, NSW 2006, Australia

Annales Polonici Mathematici, Tome 84 (2004) no. 2, pp. 147-167

Résumé

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 $M(0,5)$.

Détail

DOI : 10.4064/ap84-2-5

Keywords: hyperbolic geodesic joining punctures riemann surface has infinite length obtain useful distance like quantity define finite pseudo length geodesic terms hyperbolic length its surrounding geodesic loop there defined angle between geodesics meeting puncture pseudo trigonometry connects these angles pseudo lengths state prove theorem resembling ptolemys classical theorem cyclic quadrilaterals three general lemmas intersections shortest sense pseudo length geodesic joins these ideas applied description optimal fundamental region covering fuchsian group five punctured sphere effectively giving fundamental region modular group