Geodesics on Riemann surfaces with ramification points of order greater than two (Q5947944)

In studying the geometry of Fuchsian and Kleinian groups, it is generally possible to ignore the effects of elliptic elements by passing to a cover of finite order. This method has the disadvantage that one loses information. In this paper the author investigates some rather special cases of Fuchsian groups directly in order to gain experience with this type of question. The groups investigated are the subgroups of the modular group of index two and three, Hecke groups and a few others. The paper is to a large extent descriptive; the main result show that for the subgroup of the modular group of index 3 the number of self-intersections along a geodesic arc is essentially determined by the length of the arc. The techniques used are those of plane topology.