Crossroads between hyperbolic geometry and number theory

Abstract

The motivation of this text is to explain through two elementary examples some links between hyperbolic geometry and another domains of the mathematics. In the first example, we establish relations between the linear orbits of the discrete subgroups of SL(2,R) and the horocyclic trajectories on the hyperbolic surfaces. In the second one, we use the topology of the geodesic trajectories on the modular surface to obtain results in the theory of Diophantine approximations.