Diophantine approximation in Fuchsian groups

Abstract

This paper is concerned with the geometric and measure-theoretic structure of the limit set of a Fuchsian group. By a Fuchsian group we shall understand a finitely generated Fuchsian group; we shall not attempt to investigate the pathologies of infinitely generated Fuchsian groups. The present work splits into two parts. Up to § 7 we give a complete account of the geometry of the action of a Fuchsian group both on the open disk and on the unit circle. Although this has been studied in the past, the account given here is more detailed and systematic than anything in the literature. The detail, which at times may seem excessive, is required for applications in the second part. The other part §§8—10, adopts the following point of view. The rational numbers can be characterized as the parabolic vertices of the modular group r . The theory of diophantine approximation (see, for example, Cassels 1965) gives ways of describing how well the rationals approximate a given number. The corresponding question for a Fuchsian group is: how well do the images of a distinguished point approximate an arbitrary limit point? This problem has already been raised by (Rankin 1957) and (Lehner 1964), and to some extent answered by them. The first part of this paper contains a complete solution. In the second part we push the analogy further and seek theorems concerning the behaviour of almost all points-that is, corresponding to ‘metric number theory’. In fact we can obtain results almost (but not quite) as sharp as their classical counterparts. This is carried out in § 9 and the structure of the exceptional set is described in § 10. O f course, this is only meaningful for groups of the first kind.