L'horizon de SOL

The goal of this paper is to give an explicit analysis of the geodesic flow on the three dimensional Lie group SOL. In particular we describe its horizon. (The horizon of a riemannian manifold is a topological space parametrizing the asymptotic classes of geodesic rays.) We begin the paper by a brief exposition of some known results about the asymptotic behaviour of the geodesics in manifolds of negative and positive curvature. Sections two and three present the necessary notions of SOL geometry and the equations of the geodesics are integrated in section 4. In Section 5, we classify the geodesics in three types according to their geometric behaviour (reflecting the non isotropic character of SOL geometry) and in Section 6 we finally compute the horizon.