Computing self-intersections of closed geodesics on finite-sheeted covers of the modular surface

Source abstract

An algorithm is given for deciding whether a closed geodesic on a finite-sheeted cover of the modular surface has self-intersections; if it does, the algorithm gives them in the order they occur in traversing the geodesic. The following general result on geodesics is proved: any closed geodesic on a Riemann surface R can be lifted to a simple closed geodesic on some finite-sheeted cover of R . In the last two sections the connection with the stabilizer (under the modular group) of a Markov quadratic irrationality is discussed.