Diophantine approximation on hyperbolic Riemann surfaces

Abstract

Following the leads of H. Cohn and A. Schmidt we shall investigate geometric structures on hyperbolic Riemann surfaces for which Markoff-like theorems hold. The idea, which is similar to the approach taken by D. Sullivan in [21] is to look at the affinity of a geodesic for the noncompact end of a surface. More precisely, we consider the maximal depth a geodesic travels into a noncompact end. The spectrum of depths has the same structure as Markoff's spectrum with the correspondence given in terms of the geodesics length and topology. The upper discrete part of the spectrum is occupied by the simple closed geodesics and the lower limit value of the discrete spectrum is occupied by the geodesics that are limits of simple closed ones. The reason for this interaction between the geometry and the number theory becomes apparent when we transfer our attention to a Fuchsian group representing the hyperbolic surface. For example, consider the classical Modular group M6bz. The orbit of infinity under the action of M6bz is exactly the set of rational numbers. It is also the set of limit points which are fixed by parabolic transformations in the group. Each parabolic fixed point corresponds, in a sense, to the noncompact end on the quotient surface. As we shall see, the degree to which a number x is approximated by a rational number is directly related to the depth a geodesic with the endpoint x travels